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Symmetry
&
Symmetry breaking
© Maurits Cornelis Escher
Paolo Finelli,
Physics Department, University of Bologna,
Nuclear Physics Course - 2012
Contents
1 Symmetry and Symmetry Breaking
3
1.1
Definition of Spontaneous Symmetry Breaking . . . . . . . . . . . . . . . . . . .
5
1.2
Variational symmetry or symmetry as a unitary transformation? . . . . . . . . .
6
1.3
Coleman Theorem and Goldstone Theorem . . . . . . . . . . . . . . . . . . . . .
8
1.3.1
1.3.2
The invariance of the vacuum is the invariance of the world (Coleman
Theorem [5]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
Field theories with superconductor solutions (Goldstone Theorem [6, 7, 8])
13
2 Examples of Spontaneous Symmetry Breaking
2.1
2.2
2.3
2.4
21
Complex scalar fields: U (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.1.1
General background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.1.2
Symmetry breaking potential . . . . . . . . . . . . . . . . . . . . . . . . .
24
Higgs mechanism [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.2.1
Local gauge symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
Real scalar fields: SO(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
2.3.1
Application of the Goldstone Theorem . . . . . . . . . . . . . . . . . . . .
36
Generalization to n-Lie group . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
A History of Spontaneous Symmetry Breaking
45
B Group Theory - a short introduction
48
B.1 Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
B.2 Lie Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
B.3 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
C Wigner theorem
58
D Noether theorem
61
Chapter 1
Symmetry and Symmetry Breaking
From Ref. [1], a general definition is
The term symmetry derives from the Greek word συµµτ ρια (meaning with measure)
and originally indicated a relation of commensurability (such is the meaning codified
in Euclid’s Elements for example). It quickly acquired a further, more general,
meaning: that of a proportion relation, grounded on (integer) numbers, and with
the function of harmonizing the different elements into a unitary whole. Symmetry
is closely related to harmony.
In modern science (not only physics)
The group-theoretic notion of symmetry is the one that has proven so successful
in modern science. Symmetry remains linked to regularity and unity: by means of
the symmetry transformations, distinct (but equal or, more generally, equivalent)
elements are related to each other and to the whole, thus forming a regular unity.
The definition of symmetry as invariance under a specified group of transformations
allowed the concept to be applied much more widely, not only to spatial figures but
also to abstract objects such as mathematical expressions in particular, expressions
of physical relevance such as dynamical equations.
In particular for quantum mechanics
In general, if G is a symmetry group of a theory describing a physical system (that
is, the fundamental equations of the theory are invariant under the transformations
of G), this means that the states of the system transform into each other according
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Eigenvalue spectra
of the invariants
of the symmetry
group
Labels for
classifying the
irreducible
representations
Invariant properties
of the physical
system
E
SO(3) Casimir
invariant
L2=Lx2+Ly2+Lz2
l, angular
momentum
D (l=2)
P (l=1)
2l+1
degeneracy
S (l=0)
Example
to some representation1 of the group G. In other words, the group transformations
are mathematically represented in the state space by operations relating the states
to each other. In quantum mechanics, these operations are generally the operators
acting on the state space that correspond to the physical observables, and any state of
a physical system can be described as a superposition of states of elementary systems,
that is, of states which transform according to the irreducible representations of the
symmetry group. The observables representing the action of the symmetries of the
theory in the state space, and therefore commuting with the Hamiltonian of the
system, play the role of the conserved quantities. The eigenvalue spectra of the
invariants of the symmetry group provide the labels for classifying the irreducible
representations of the group: on this fact is grounded the possibility of associating
the values of the invariant properties characterizing physical systems with the labels
of the irreducible representations of symmetry groups, i.e. of classifying elementary
physical systems by studying the irreducible representations of the symmetry groups.
1
Group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented
by matrix multiplication. They describe how the symmetry group of a physical system affects the solutions of
equations describing that system.
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1.1
Corso di Teoria delle Forze Nucleari (2012)
Definition of Spontaneous Symmetry Breaking
Here we collect two definitions of Spontaneous Symmetry Breaking (SSB)2 :
1. Generally, the breaking of a certain symmetry does not imply that no symmetry is present,
but rather that the situation where this symmetry is broken is characterized by a lower
symmetry. In group theoretic terms, this means that the initial symmetry group
is broken to one of its subgroups. It is therefore possible to describe symmetry
breaking in terms of relations between transformation groups, in particular between a
group (the unbroken symmetry group) and its subgroup(s) [1]
2. Spontaneous symmetry breaking (SSB) indicates a situation where, given a symmetry of
the equations of motion, solutions exist which are not invariant under the action
of this symmetry without the introduction of any term explicitly breaking the
symmetry (whence the attribute spontaneous). When some parameter (order parameter)
reaches a critical value, the lowest energy solution respecting the symmetry of the theory
ceases to be stable under small perturbations and new asymmetric (but stable) lowest
energy solutions appear. The new lowest energy solutions are asymmetric but are all
related through the action of the symmetry transformations. In other words, there is
a degeneracy (infinite or finite depending on whether the symmetry is continuous or
discrete) of distinct asymmetric solutions of identical (lowest) energy, the whole
set of which maintains the symmetry of the theory. SSB occurs both in classical
and in quantum physics3 [1].
For history and relevant bibliography in Quantum Field Theory, please see Appendix A.
2
The adjective spontaneous differentiates symmetry breaking that arises due to the noninvariance of the
vacuum state from that due to explicitly adding asymmetric terms to the Lagrangian.
3
A distinction has to be drawn is between finite and infinite physical systems. In the case of finite systems,
SSB actually does not occur: tunnelling takes place between the various degenerate states, and the true lowest
energy state or ground state turns out to be a unique linear superposition of the degenerate states. In fact, SSB is
applicable only to infinite systems - many-body systems (such as ferromagnets, superfluids and superconductors)
and fields - the alternative degenerate ground states being all orthogonal to each other in the infinite volume
limit and therefore separated by a superselection rule. A superselection rule is a contraction of the Hilbert space.
This means that the Hilbert space of the system is built up from one of the ground-states |Ωi, and other Hilbert
spaces built on other ground-states become inaccessible because there are no local observables that can connect
them [4].
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1.2
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Variational symmetry or symmetry as a unitary transformation?
A typical definition of SSB is that the vacuum state of a broken symmetry theory is not invariant
under all the symmetries of the underlying Lagrangian. Symmetry breaking results from a
mismatch between a) variational symmetries of the Lagrangian and b) symmetries that can be
defined as unitary transformations on the Hilbert space of states. The second sense of symmetry
is familiar in quantum mechanics: a symmetry transformation preserves transition probabilities;
that is, it is an (invertible) map f : |φi → |φ0 i defined on states |φi in a Hilbert space such that
for all φ and ψ, |hφ|ψi| = |hφ0 |ψ 0 i|. Wigner proved that corresponding to any such mapping
f there is a linear and unitary (or antilinear and antiunitary4 ) operator Û implementing the
symmetry transformation. The mismatch between the two senses of symmetry, a) and b), occurs
when there is no unitary operator corresponding to the Noether charge generating a variational
symmetry.
Noether’s first theorem establishes the existence of a conserved charge for every global variational symmetry of the Lagrangian5 . The theorem applies to the broad class of theories that
R
derive equations of motion via Hamilton’s principle from the action S = R d4 x L(φ, ∂µ φ, xµ )
where φ(x) are the dependent variables, xµ are the coordinates, and the Lagrangian density L
is integrated over a compact space-time region R. A solution φ(xµ ) is a map from space-time
to the space of field variables such that the equations of motion, the Euler-Lagrange equations
for L, are satisfied. Suppose that there is an r−parameter Lie group G whose elements map
(x, φ) → (x, φ0 ) such that S is invariant. Noether’s first theorem establishes that then there are
r−conserved currents J µ (φ) such that ∂µ J µ (φ) = 0. The charge associated with the symmetry
R
is the integral of the time component of this conserved current, that is, Q(φ) = R d3 x J 0 ; it fol-
lows from the vanishing divergence of the four vector that Q(φ) is constant and that dQ/dt = 0,
if the current flux vanishes on the boundary of the region R. If the two senses of symmetry have
to be matched, then in the quantized field theory based on this Lagrangian one would find a
4
Antiunitary operators correspond to symmetries that are not continuously connected to the identity, such as
time reversal. Unitary and linear operators have to satisfy the following relations
(U φ, U ψ)
=
(φ, ψ)
(1.1)
U (ξφ + ηψ)
=
ξU φ + ηU ψ ,
(1.2)
whereas antilinear and antiunitary operators satisfy
(U φ, U ψ)
U (ξφ + ηψ)
(φ, ψ)∗
=
∗
=
(1.3)
∗
ξ Uφ + η Uψ ,
(1.4)
where (·, ·) is the usual internal product. See Weinberg (1995, Sect 2.6, Vol. 1) for more details about Wigner’s
theorem or Appendix C.
5
In the following we will consider only internal (no space-time) symmetries, see Appendix D for more details.
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Symmetries as
Unitary transformations
(Wigner Theorem)
Variational symmetries
Noether Theorem
�
�
�φ |ψ � = �φ|ψ�
Conserved
charges
�
Q = d3 J0 (x)
U = eiχQ
SSB
but the set of
degenerate vacua
respects the
symmetry of L
�
|Φ0 �
|Φ0 �
U = eiχQ
Figure 1.1: Realization of a spontaneously broken symmetry.
one-parameter family of unitary operators Û (ξ) = eiξQ̂ implementing the symmetry, where Q̂ is
the operator corresponding to the Noether charge. It can be showed that if the vacuum state
|0i is translationally invariant, then the vacuum is either invariant under the internal symmetry,
Q̂|0i = 0, or there is no state corresponding to Q̂|0i in the Hilbert space.
Fabri-Picasso theorem. There are only two possibilities:
1. Q̂|0i = 0 and |0i is an eigenstate of Q̂ with eigenvalue 0, so that |0i is invariant under Û
(i.e. Û |0i = |0i).
2. @ Q̂|0i in the space (its norm is infinite). This statement is more accurate than more
intuitive statements like Q̂|0i =
6 0 widely used in literature.
By definition, an internal symmetry implemented by Q̂ commutes with the four-momentum
operators P̂ µ , i.e. [Q̂, P̂ µ ] = 0, and, by translation invariance of the vacuum state, eiP̂ ·x |0i = |0i.
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These two facts imply that
h0|J0 (x)Q̂|0i = h0|eiP̂ ·x J0 (0)e−iP̂ ·x Q̂|0i = h0|J0 (0)Q̂|0i .
(1.5)
The norm of Q̂V can then be calculated by integrating the current
h0|Q̂Q̂|0i =
Z
V
3
d x h0|J0 (x)Q̂|0i =
Z
V
d3 x h0|J0 (0)Q̂|0i ,
(1.6)
which diverges as V → ∞ unless Q̂|0i = 0 (Ref. [2] and Ref. [3] pgs. 197-198). The second case
corresponds to SSB. The symmetry is hidden in that there is no unitary operator to
map a physical state to its symmetric counterparts; instead, the symmetry is (roughly
speaking) a map from one Hilbert space of states to an entirely distinct space. This is usually
described as vacuum degeneracy although each distinct Hilbert space has a unique
vacuum state.
1.3
Coleman Theorem and Goldstone Theorem
Let brief review some basic results [4]. We start from a Lagrangian density (of real scalar fields
for simplicity)6
L (φi (x), ∂µ φi (x)) ,
(1.7)
which gives us by the principle of minimal action
δ
Z
d4 x L = 0
(1.8)
the Euler-Lagrange equations of motion
∂µ
∂L
∂(∂µ φi )
−
∂L
=0.
∂φi
(1.9)
If the Lagrangian (1.7) is invariant under an n parametric transformation group
φi (x) → Vij (ξ1 , ξ2 , . . . , ξn )φj (x)
V
=
eiξk Ik ,
(1.10)
(1.11)
6
From this point onwards we omit the hat (ˆ) over the operators whenever is possible for simplicity, to ease
readability of the text.
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where Ik are the infinitesimal generators, the equations of motion (1.9) will be automatically
invariant respect to the same transformations (1.10). At the same time, Noether’s theorem gives
us n conserved currents
Jkµ = −i
∂L
Ik φ
∂µ φ
(1.12)
(in matrix notation). Since the momentum canonically conjugate to φ is
Π=
∂L
,
∂(∂0 φ)
(1.13)
satisfying the well-known Poisson bracket relation7
[φ(x), Π(y)] = δ(x − y)
(1.15)
[φ(x), φ(y)] = 0
(1.16)
[Π(x), Π(y)] = 0 ,
(1.17)
we have, considering that Jk0 = −iΠIk φ,
Jk0 (x), φ(y) = iIk φ(x)δ(x − y)
0
Jk (x), φi (y) = i (Ik )ij φj (x)δ(x − y)
(1.18)
(matrix notation)
(1.19)
or, introducing the conserved charges,
Qk =
Z
d3 x Jk0 (x)
[Qk , φi (y)] = i (Ik )ij φj (y) .
we have
(1.20)
(1.21)
Qk is therefore the generator of the infinitesimal canonical transformations corresponding to
(1.10). It is natural to introduce an operator
U (ξ) = eiξk Qk ,
(1.22)
7
In canonical coordinates (qi , pj ), on the phase space, given two functions f (pi , qi , t), and g(pi , qi , t), the
Poisson bracket takes the form
N X
∂f ∂g
∂f ∂g
[f, g] =
−
.
(1.14)
∂qi ∂pi
∂pi ∂qi
i=1
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in order to implement unitarily the symmetry8
U (ξ)φi (x)U −1 (ξ) = Vij φj (x) .
(1.23)
In fact Eq. (1.21) could also be obtained expanding (1.23) in the exponential representation.
Let consider a unitary operator inducing the group tranformation
U = eiξk Qk ,
where Qk are the conserved charges (or the group generators associated to the k Noether currents). If the vacuum9 is invariant under the group transformation (i.e. U |0i = |0i) then it is
necessarily a singlet10 and it is annihilated by the symmetry generators, namely
Qk |0i = 0 .
(1.24)
In fact if we consider the infinitesimal transformation U (ξk )|0i = (1 + iξk Qk )|0i = |0i, it imme-
diately follows that the group generator Qk has the property to annihilate the vacuum (1.24).
This is the so called Wigner-Weyl realization of the symmetry. The Hamiltonian H can be
shown to remain invariant with respect to continuous transformations generated by the group
G and the symmetry manifests itself directly in the spectrum of H as degenerate multiplets.
The Wigner-Weyl realization is a sort of accounting symmetry since, for example, it allows to
classify the particles according to the irreducible representations of the group G (like the isospin
label for baryon multiplets in the hadron spectrum). It is easy to see that multiplet structures
emerge naturally if the vacuum is left invariant under the symmetry transformation. To prove
the last sentence, let consider two states |Ai and |Bi:
|Ai = φ†A |0i ,
|Bi = φ†B |0i ,
(1.25)
where φ†A and φ†B are supposed to relate to each other by a vector transformation
[Q, φ†A ] = φ†B
(1.26)
8
For every element g ∈ G, it is possible to, given any representation T in the space L, define a new represen0
tation TU (g) acting in the vector space L as follows
TU (g) = U T (g)U −1 .
T (g) and TU (g) are equivalent representations. We define, as usual, the group generators as Ik the derivatives of
the operator T (ξk ) respect to the parameter ξk , taken at ξk = 0.
9
We define the vacuum |0i as the state of the system for which h0|H|0i = min.
10
A singlet is a one-dimensional representation.
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for some generator Q of the symmetry group, such that
[Q, H] = 0 .
(1.27)
U φ†A U −1 ' φ†A + iφ†B
(1.28)
Eq. (1.26) is equivalent to
for an infinitesimal transformation U ' 1 + iQ. Thus φ†A is rotated into φ†B by U , and the
operators will create states related by the symmetry transformation. Let’s assume that
H|Ai = EA |Ai H|Bi = EB |Bi ,
(1.29)
what assumption is necessary to prove that EA = EB is satisfied? We have
EB |Bi = H|Bi = Hφ†B |0i = H(Qφ†A − φ†A Q)|0i .
(1.30)
Now if Q|0i = 0 we can rewrite the right-hand side of the previous equation as follows
HQφ†A |0i = QHφ†A |0i
= QH|Ai = EA Q|Ai
= EA Qφ†A |0i = EA (φ†B + φ†A Q)|0i
= EA |Bi .
(1.31)
If Q|0i = 0 then follows EA = EB , and multiplets appear naturally in the energy spectrum.
1.3.1
The invariance of the vacuum is the invariance of the world (Coleman
Theorem [5])
If a generator Qa of a continuous symmetry group G is given as a space integral of some current
density Jaµ (x, t), and if it has the property to annihilate the vacuum (so the vacuum is invariant
under G), then the Hamiltonian remains invariant under transformations of the fields according
to G and the current is conserved.
Proof: If the vacuum is invariant under the group then the generator of the group must annihilate the vacuum. That is to say,
Qa (t)|0i =
Z
d3 x Ja0 (x, t)|0i = 0 ,
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then
hn|
Z
3
d x
Ja0 (x, t)|0i
=
Z
d3 x hn|Ja0 (x, t)|0i = 0 ,
is certainly correct for any arbitrary state |ni, and, of course, for a state |ni with vanishing
3-momentum (p = 0) and non-zero energy11 (p0 6= 0). If the previous relation is valid then12
hn|Ja0 (x)|0i = 0 x = (x, t)
is also valid (because of the way we have chosen the momentum), which is the same as
hn|∂µ Jaµ (x)|0i = 0 .
(1.32)
Lorentz-invariance tells us that if Eq. (1.32) is true in one Lorentz frame, it is true in all Lorentz
frames. Since any momentum eigenstate can be obtained by applying a Lorentz transformation
to a state with zero 3-momentum, the latter equation is true for any momentum state on the
left. This is to say that
∂µ J µ (x)|0i = 0 .
In QFT there is a theorem (by Federbush and Johnson) which states that any local operator13
which annihilates the vacuum vanishes identically. Therefore
∂µ J µ = 0 ,
as it should be from Noether’s theorem. This implies that the generator Qa (t) is independent
of time and commute with the Hamiltonian H,
dQa
= i[Qa , H] = 0.
dt
If the vacuum is not invariant under the symmetry operation associated with m ≤ n genera-
tors Qa , then the corresponding symmetry operation applied to the vacuum leads to new states
so that
Qa |0i =
6 0
or, better ||Qa |0i|| = ∞ .
(1.33)
11
In fact non zero energy modes are consequences of a spontaneously broken realization of the symmetry.
It is easy to prove that the integral relation exists, at least with a dense set of states on the left. On the
other hand if we work with momentum eigenstates this proof can not be considered rigorous, see [7].
13
Charges are not local operator because Qa is defined as an integral over the space.
12
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In the spectrum of the Hamiltonian there necessarily exists a branch of elementary gap-free
excitations whose energies go to zero as the momentum goes to zero. This type of Hamiltonian
invariance does not imply the existence of multiplet structure; we call it a Nambu-Goldstone
realization of the symmetry. In cases in which the symmetry is spontaneously broken, the total
charge of the conserved Noether current associated with the symmetry transformation is not
identical to the generator of the corresponding unitary group. Such charges/generators are
known as broken charges. In the following we will show that in this case the total charges
indeed do not exist as Hermitian operators in a Hilbert space and that the states
of the system do not transform according to a irreducible representation of the
symmetry group. The phenomenon of vacuum non-invariance can be explained assuming the
existence of some non-vanishing macroscopic averages of local operators in the ground-state (so
called anomalous averages or order parameters). Each value of an anomalous average will then
define a unique vacuum and a corresponding Hilbert space.
1.3.2
Field theories with superconductor solutions (Goldstone Theorem [6,
7, 8])
A spontaneously broken symmetry realization is identified by systems in which the ground state
is not an eigenstate of some generators of the global symmetry of the Hamiltonian. Given the
charge density J 0 (x), one introduces for an arbitrary finite space domain Ω the operator
QΩ (t) =
Z
d3 x J 0 (x, t)
(1.34)
Ω
The symmetry breaking condition can be restated as the existence of a (not necessarily local)
operator Φ such that
lim h0|[QΩ (t), Φ]|0i =
6 0
Ω→∞
(1.35)
where |0i is a translationally invariant ground state. This expectation value is known as the
order parameter. Clearly, this formal definition immediately implies the previous one: if the
vacuum were an eigenstate of the charge operator, the expectation value of this commutator
would have to be zero. It is customary to identify Q(t) = limΩ→∞ QΩ (t) formally with the
integral charge operator. However, this operator strictly speaking does not exist because of the
Fabri-Picasso theorem. The intuitive picture of spontaneous symmetry breaking, based on the
observation that a symmetry transformation does not leave the ground state intact, suggests
high degeneracy of equivalent ground states. Indeed, since the charge operator commutes with
the Hamiltonian, so will a finite symmetry transformation generated by this operator. It will
therefore transform the ground state into another state with the same energy. As long as the
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symmetry group is continuous, we will find infinitely many degenerate ground states. On account
of the fact that they are all connected by symmetry transformations, they must be physically
equivalent and any one of them can serve as a starting point for the construction of the spectrum
of excited states.
The finite volume charge operator QΩ (t) induces a finite symmetry transformation, UΩ (θ, t) =
exp[iθQΩ (t)], which in turn gives rise to a rotated ground state, |θ, tiΩ = UΩ† (θ, t)|0i. However,
like the limit limΩ→∞ QΩ (t) does not exist, the operator exp[iθQ(t)] is not well defined either.
In fact, it can be proved that
lim h0|θ, tiΩ = lim h0| exp[−iθQΩ (t)]|0i = 0 .
Ω→∞
(1.36)
Ω→∞
It means that in the infinite volume (thermodynamic) limit, any two ground states, formally
connected by a symmetry transformation, are actually orthogonal. The same conclusion holds for
excited states constructed above these vacua. All these states therefore cannot be accommodated
in a single separable Hilbert space, forming rather two distinct Hilbert spaces of their own. Any
of these Hilbert spaces can, nevertheless, be taken as a basis for an equivalent description of
the system, and the choice has no observable physical consequences. Unlike the transformations
of physical states, finite symmetry transformations of observables can be consistently defined.
Using the Baker–Campbell–Hausdorff formula one obtains for any operator A that
1
Aθ,t;Ω ≡ UΩ (θ, t)AUΩ† (θ, t) = A + iθ[QΩ (t), A] + (iθ)2 [QΩ (t), [QΩ (t), A]] + . . .
2
where
[QΩ (t), A] =
Z
d3 x [J 0 (x, t), A] .
(1.37)
(1.38)
Ω
As long as the theory satisfies the microcausality condition14 , that is, the commutator of any
two local operators separated by a spacelike interval vanishes, and as long as the operator A is
localized in a finite domain of spacetime, there will be a region Ω0 such that the charge density
14
The requirement that the causality condition (which states that cause must precede effect) be satisfied
down to an arbitrarily small distance and time interval. The microcausality condition usually refers to distances
≤ 10−14 cm and to times ≤ 10−24 sec. It is shown in the theory of relativity that the assumption of the existence of
physical signals that propagate with a velocity greater than the velocity of light leads to violation of the causality
requirement. Thus, the microcausality condition prohibits the propagation of signals at a velocity greater than
the velocity of light in the small. In quantum theory, where operators correspond to physical quantities, the
microcausality condition requires the interchangeability of any operators that pertain to two points of space-time
if these points cannot be linked by a light signal. This interchangeability means that the physical quantities to
which these operators correspond can be precisely determined independently and simultaneously. The violation of
the microcausality condition would make it necessary to radically alter the method of describing physical processes
and to reject the dynamic description used in modern theories, in which the state of a physical system at a given
moment of time (the effect) is determined by the states of the system at preceding times (the cause). [Grigor0 ev]
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outside this region does not contribute to the commutator,
Z
3
0
d x [J (x, t), A] = 0
and
R3 \Ω0
lim [QΩ (t), A] =
Ω→∞
Z
d3 x [J 0 (x, t), A]
(1.39)
Ω0
The transformation UΩ (θ, t)AUΩ† (θ, t) therefore has a well-defined limit as Ω → ∞. The expectation value of the rotated operator Aθ,t;Ω in the vacuum |0i can then be interpreted as the
expectation value of A in the rotated vacuum |θ, tiΩ .
Goldstone theorem can be proved under the following basic hypotheses:
1. The degenerate vacuum is invariant under a subgroup F of the symmetry group of the
hamiltonian G.
2. Lorentz covariance of the theory (in non-relativistic approaches the theorem still applies
but with non-trivial consequences about the counting of the Nambu-Goldstone modes).
Under the previous hypotheses it follows the Goldstone theorem15 :
For each broken symmetry (each broken charge) one massless mode
(massless particle) appears in the energy spectrum.
R
Given QΩ (x0 ) = Ω d3 x J 0 (x) where Ω is the volume, if
lim [H, QΩ ] = 0
Ω→∞
and
lim ||QΩ |0i|| = ∞
Ω→∞
and if an operator A exists with
lim h0|[QΩ (x0 ), A]|0i 6= 0
|
{z
}
order parameter
Ω→∞
then a massless excitation (particle) is present in the energy spectrum.
We recall here a standard proof of the Goldstone theorem. We start showing that if a Lagrangian
is invariant under a symmetry transformation then a current is conserved
∂µ J µ (x) = 0 ,
(1.40)
15
There are exceptions, the most important is the case of gauge theories. For any sponteously broken local
symmetry, one Goldstone boson disappears from the physical spectrum of the states and the corresponding gauge
bosons acquires a mass (due to the fact that is impossible to mantain at the same time manifest Lorentz-covariance
and positivity of the Hilbert space), the so called Higgs mechanism (see Sect. 2.2).
15
[23/04/2012]
Paolo Finelli
Corso di Teoria delle Forze Nucleari (2012)
and for a local observable A (no assumptions about it, except that it is localized in a finite
region of space)
lim
Ω→∞
d
QΩ (t), A = 0
dt
(1.41)
where
QΩ (t) =
Z
d3 x J 0 (x, t) .
(1.42)
Ω
In fact, we have (applying the Gauss law)
Z
d
QΩ (t), A +
dS · [J (x, t), A] .
d x [∂µ J (x, t), A] =
0=
dt
S
Ω
Z
3
µ
(1.43)
For Ω → ∞ the surface integral vanishes because the fields are supposed to vanish at the
boundaries (a common prescription in QFT). This is equivalent to say that
lim [QΩ (t), A] = Φ
Ω→∞
where Φ is time-indipendent
dΦ
=0
dt
and
Φ = Φ(r).
The Nambu-Goldstone mechanism is realized if16
h0|Φ(r)|0i =
6 0,
where |0i is a traslationally invariant vacuum state. This relation implies that |0i cannot be an
eigenstate of Q, and it follows from exp(iξQ)|0i =
6 |0i that the corresponding operator U is not
a unitary operator. Please note that in the space of the eigenvectors of the observable A the
charge Qa is unobservable because it does not commute with A.
Let consider now a set of local operators φi (x) not invariant under a continuous symmetry
R
generated by the charge QaΩ = Ω d3 x J0a , then, by definition,
lim h0|[QaΩ (t), φi (x)]|0i =
6 0.
Ω→∞
(1.44)
16
More precisely the anomalous average of Φ(r) corresponding to a given Hilbert space is defined as an average
over the volume Ω in the following way
Z
Z
1
1
h0| lim
d3 r Φ(r)|0i = lim
d3 r h0|Φ(r)|0i 6= 0 .
Ω→∞ Ω Ω
Ω→∞ Ω Ω
16
[23/04/2012]
Paolo Finelli
Corso di Teoria delle Forze Nucleari (2012)
Explicitly, we have
h0|[QaΩ (t), φi (x)]|0i
= h0|
Z
3
d y
Ω
and inserting a density of states 1 =
→
XZ
Ω
n
P
[J0a (y), φi (x)]|0i
=
Z
Ω
d3 y h0|[J0a (y), φi (x)]|0i ,
n |nihn|:
d3 y (h0|J0a (y)|nihn|φi (x)|0i − h0|φi (x)|nihn|J0a (y)|0i) .
(1.45)
Then we make use of translational invariance17
J0a (y) = eipy J0a (0)e−ipy
to obtain from Eq. (1.45)
→
XZ
Ω
n
d3 y h0|eipy J0a (0)e−ipy |nihn|φi (x)|0i − h0|φi (x)|nihn|eipy J0a (0)e−ipy |0i .
Evaluation of the action of the momentum operator p over the states |ni (and including explicitly
the limit Ω → ∞) gives
lim
Ω→∞
XZ
n
Ω
h
i
d3 y h0|J0a (0)|nihn|φi (x)|0ie−ipn y − h0|φi (x)|nihn|J0a (0)|0ieipn y 6= 0 .
Performing the spatial integration (we now take safely the limit Ω → ∞)
Z
0
d3 y e−ipn y = (2π)3 δ 3 (pn )e−iEn y = (2π)3 δ 3 (pn )e−iEn t
Ω
we have
X
n
h
i
(2π)3 δ 3 (pn ) h0|J0a (0)|nihn|φi (x)|0ie−iEn t − h0|φi (x)|nihn|J0a (0)|0ieiEn t
= h0|Φ(r)|0i = v ∈ C .
Now this equation is valid for all times t and, since we have shown that Φ(r) does not depend
on t, it follows that the left-hand side of this equation must not depend on time. Clearly these
conditions are consistent only if the left-hand side vanishes except for those states |ni where
h0|φi (x)|nihn|J0a (0)|0i =
6 0
17
for En |pn →0 = 0
(1.46)
Assuming that eipx |0i = |0i and keeping in mind that the operators are local.
17
[23/04/2012]
Paolo Finelli
Corso di Teoria delle Forze Nucleari (2012)
It means that only those states |ni for which the energy vanishes as the 3-momentum goes to
zero
p→0
En = 0 m2n = p2 = 0 ,
contribute. In relativistic field theories this implies the existence of massless particles. The state
|ni must have the same quantum number as φi (y) and J0a (x). (In particular this state must
have the same Lorentz properties of the charge Qa ). The Goldstone particles have to be of
zero energy since there is no accompanying change in the energy of the system in going from
one vacuum to the other.
One can also demonstrate the opposite of the Goldstone theorem: if H does not have a massless
particle in its spectrum, the operator
U (η) = lim exp iη
Ω→∞
Z
is unitary.
Paolo
PaoloFinelli
Finelli
Ω
d x J (x) = lim UΩ (η, t)
3
0
Ω→∞
Corso
Corso di
di Teoria
Teoria delle
delle Forze
Forze Nucleari
Nucleari (2011)
(2011)
Wigner-Weyl
realization
Wigner-Weyl
Wigner-Weyl
realization
realization
Nambu-Goldstone
realization
Nambu-Goldstone
Nambu-Goldstone
realization
realization
Exact symmetry
Spontaneous symmetry
breaking
Exact
Exact symmetry
symmetry
Q|0�
Q|0�==00
Degenerate multiplets
multiplets
EDegenerate
Spontaneous
Spontaneous symmetry
symmetry breaking
breaking
||Q|0�||
||Q|0�||==∞
∞
Massless
Massless
Goldstone
Goldstone bosons
bosons
E
Mass gap
degenerate
multiplets
Massless
mode as
ItIt means
means that
that only
only those
those states
states |n�
|n� for
for which
which the
the energy
energy vanishes
vanishes
as the
the 3-momentum
3-momentum
goes
goesto
tozero
zero
0
Some general remarks.
0
pp→
→00 EEnn ==00 m
m2n2n ==pp22 ==00,,
contribute.
contribute.
In
In equations
relativistic
relativistic
field
fieldand
theories
theories
this
this
implies
implies
the
theofexistence
existence
of
ofatmassless
massless
particles.
1) The
of motion
the currents
involve
products
field operators
the same particles.
point
and therefore
are ill
defined
quantities whose
properas
meaning
by particular
The
The state
state |n�
|n�
must
must
have
have the
the
same
same
quantum
quantum
number
number
as
φφii(y)
(y)should
and
and be
JJ0a0aobtained
(x).
(x). (In
(In
particular
limiting procedures starting from different space-time points.
this
this state
state must
must have
have the
the same
same Lorentz
Lorentz properties
properties of
of the
the charge
charge Q
Qaa).). The
The Goldstone
Goldstone
particles
particles have
have to
to be
be of
of zero
zero energy
energy since
since there
there isis no
no accompanying
accompanying
change
change in
in the
the
18
[23/04/2012]
energy
energyofofthe
thesystem
system in
in going
going from
from one
one vacuum
vacuum to
to the
the other.
other.
One
One can
can also
also demonstrate
demonstrate the
the opposite
opposite of
of the
the Goldstone
Goldstone theorem:
theorem: ifif H
H does
does not
not have
have aa
Paolo Finelli
Corso di Teoria delle Forze Nucleari (2012)
2) The construction of the charge from
R
d3 x J 0 requires in the classical physics case the
hypothesis that the fields should vanish at infinity to ensure convergence of the integral.
This is physically reasonable. In the quantum case the existence of vacuum fluctuations
occuring all over space (translation invariance) does not allow us to take the quantum
analogue of the charge as a well defined operator even if a meaning has been given to the
density. For this reason one must work with the vacuum expectation values.
3) Spontaneous breaking of a global continuous symmetry implies the existence of poles at
p2 = 0 in certain Green’s functions (this poles are related to the presence of massless
scalar particles in the physical spectrum, of course). Let us consider the following Green’s
function
Gaµ,k (x − y) = h0|T Jµa (x)φk (y)|0i ,
(1.47)
where Jµa is the current corresponding to a generator Qa of the symmetry G and φk
belongs to an irreducible multiplet of real scalar fields. This Green’s function satisfies a
Ward identity that can be obtained by differentiating it (being careful with the derivative
of the functions involved in the time ordering)
µ
a
φj (x)δ(x − y)|0i
∂(x)
Gaµ,k (x − y) = δ(x0 − y 0 )h0|[J0a (x), φk (y)]|0i = δ(x0 − y 0 )h0| − Tkj
a
a
h0|φj (0)|0i
h0|φj (y)|0i = −δ(x − y)Tkj
= −δ(x − y)Tkj
(1.48)
We have used the transformation properties of the field as generated by the Noether current
and translational invariance of the vacuum. The Fourier transform of the Ward identity
reads
a
ipµ G̃aµ,k (p) = Tkj
h0|φj (0)|0i ,
where
Gaµ,k (x
− y) =
Z
d4 p
exp (−ip(x − y)) G̃aµ,k (p) .
(2π)4
(1.49)
(1.50)
Plugging the most general form of the (Fourier-transformed) Green’s function as allowed
by Lorentz invariance, G̃aµ,k (p) = pµ Fka (p2 ), in the Ward’s identity we get
Fka (p2 ) = −
i a
T h0|φj (0)|0i ,
p2 kj
(1.51)
which implies that the Green’s function corresponding to a generator that does not annihilate the vacuum,
a
Tkj
h0|φj (0)|0i =
6 0,
19
(1.52)
[23/04/2012]
Paolo Finelli
Corso di Teoria delle Forze Nucleari (2012)
has a pole at p2 = 0. Let us now consider the following matrix element
h0|Jµa (x)|π k (p)i = ifka pµ e−ipx ,
(1.53)
where |π k (p)i describes a particle of mass mk which is a quantum of the field φk . The
reduction formula relates this matrix element to the Green’s function G̃sµ,k (p). Defining
Gkk0
=
=
Z
Z
d4 q
−δkk0
e−iq(x−y)
4
2
(2π) q − m2k + i
d4 y G−1 (x − y)G(y − x) = δ(x − z) ,
(1.54)
we get
h0|Jµa (x)|π k (p)i =
=
Z
d4 y d4 z e−ipz Gaµ,k0 (x − y)iG−1
kk0 (y − z)
lim e−ipx G̃aµ,k0 (p)iG̃−1
kk0 (p)
p2 →m2k
= − lim e−ipx G̃aµ,k0 (p)i(p2 − m2k ) .
p2 →m2k
(1.55)
Putting together Eq. (1.53) with this equation we get,
lim G̃aµ,k0 (p)i(p2 − m2k ) = −fka pµ ,
p2 →m2k
(1.56)
which implies mk = 0, for those Green’s functions that have a massless pole (i.e. those
corresponding to generators that don’t annihilate the vacuum), as we wanted to prove.
This proof also gives us the value of fka ,
a
fka = iTkj
h0|φj (0)|0i .
(1.57)
a h0|φ (0)|0i, corresponding to
Thus, there must be a massless boson, |Πa (p)i = i|π k (p)iTkj
j
each broken generator.
For more details, see Ref. [4].
20
[23/04/2012]
Chapter 2
Examples of Spontaneous Symmetry
Breaking
2.1
2.1.1
Complex scalar fields: U (1)
General background
Let us start with a complex scalar field Φ, described by a free Lagrangian
1
L = (∂µ Φ∗ )(∂ µ Φ) − M 2 Φ∗ Φ ,
2
(2.1)
that can be interpreted as composed by two real fields Φ1 and Φ2
1
Φ∗ = √ (Φ1 + iΦ2 )
2
1
Φ = √ (Φ1 − iΦ2 ) ,
2
(2.2)
described by a free Lagrangian of two fields with the same mass M
1
1
1
1
L = (∂µ Φ1 )(∂ µ Φ1 ) − M 2 Φ21 + (∂µ Φ2 )(∂ µ Φ2 ) − M 2 Φ22 .
2
2
2
2
(2.3)
Φ as a field operator (Φ̂) has the following expansion [13]
Φ̂ =
Z
d3 k
√
â(k)e−ik·x + b̂† (k)eik·x ,
(2π)3 2ω
(2.4)
where the creation/destruction operators are defined as
â(k) =
1
√ (â1 − iâ2 )
2
21
(2.5)
Paolo Finelli
Corso di Teoria delle Forze Nucleari (2012)
b̂† (k) =
and ω =
√
1
√ (â†1 − iâ†2 ) ,
2
(2.6)
M 2 + k2 . The operators â, ↠, b̂, b̂† obey the commutation relations
h
i
0
0
â(k), ↠(k ) = (2π)3 δ 3 (k − k )
h
i
0
0
b̂(k), b̂† (k ) = (2π)3 δ 3 (k − k ) ,
(2.7)
(2.8)
while all other commutators are vanishing. The Hamiltonian operator for the complex field Φ̂
is (dropping the zero point energy, i.e. normal ordering)
Ĥ =
while the Lagrangian is
Z
i
d3 k h †
†
â
(k)â(k)
+
b̂
(k)
b̂(k)
ω,
(2π)3
1
L̂ = (∂µ Φ̂† )(∂ µ Φ̂) − M 2 Φ̂† Φ̂ .
2
(2.9)
(2.10)
In the following we summarize some basic results (details can be easily found in any quantum field
theory textbook). The classical real Lagrangian is symmetric respect to O(2) transformations1 ,
leading to a conserved (Noether) current
N µ = Φ1 ∂ µ Φ2 − Φ2 ∂ µ Φ1 ,
(2.15)
∂µ N µ = 0 ,
(2.16)
that satisfy
and a charge (constant of motion)
NΦ =
1
Z
d3 x N 0 ,
(2.17)
Rotation of coordinates about a predefined z-axis
0
φ1
0
φ2
=
(cos α)φ1 − (sin α)φ2
(2.11)
=
(sin α)φ1 + (cos α)φ2 ,
(2.12)
of an angle α, a real parameter. This is like a rotation of coordinates about the z- axis of ordinary space, but of
course it mixes field degrees of freedom, not spatial coordinates. For operators we obtain
0
φ̂1
0
φ̂2
=
(cos α)φ̂1 − (sin α)φ̂2
(2.13)
=
(sin α)φ̂1 + (cos α)φ̂2 .
(2.14)
22
[23/04/2012]
Paolo Finelli
Corso di Teoria delle Forze Nucleari (2012)
i.e. with the property that
d
NΦ = 0 ,
dt
[NΦ , H] = 0 .
(2.18)
NΦ distinguishes particles from antiparticles. In fact, in terms of Φ, we have
N µ = i(Φ∗ ∂ µ Φ − Φ∂ µ Φ∗ ) ,
(2.19)
and the symmetry (operator) charge
N̂Φ =
Z
3
0
d x N̂ =
Z
i
d3 k h †
†
â
(k)â(k)
−
b̂
(k)
b̂(k)
.
(2π)3
(2.20)
In fact N̂Φ |0i = 0. From Eqs. (2.7, 2.8) some useful relations can be derived
h
h
N̂Φ , Φ̂
N̂Φ , Φ̂†
i
i
= −Φ̂
(2.21)
= Φ̂† ,
(2.22)
and, defining Û (α) = eiαN̂Φ and expanding the exponential, we obtain the transformation law
0
Û (α)Φ̂Û −1 (α) = e−iα Φ̂ = Φ̂ ,
(2.23)
i.e., a U (1) rotation2 , a simple phase change. Consider now a state |NΦ i which is an eigenstate
of N̂Φ with eigenvalue nφ . It is easy to show that
N̂Φ Φ̂|NΦ i = (nΦ − 1)Φ̂|NΦ i ,
(2.25)
2
U (1) is the group of complex vectors of unit length. The elements of this group,
g ∈ U (1), have the form g = eiα . They form a group in the sense that
R
1. this set it is closed under complex multiplication i.e.
g = eiα ∈ U (1)
and
g 0 = eiβ ∈ U (1) → g · g 0 = eiα+β ∈ U (1)
(2.24)
S1
2. there is an identity element, i. e. g = 1
3. for every element g = eiα there is an unique inverse element g −1 = e−iα .
The elements of the group U (1) are in one-to-one correspondence with the points
of the unit circle S1 . Consequently, the parameter α that labels the transformation
(or element of this group) is defined modulo 2π, and it should be restricted to the
interval (0, 2π]. However, transformations infinitesimally close to the identity 1 lie
essentially on the straight line tangent to the circle at 1 and are isomorphic to the
group of real numbers R. The group U (1), which is compact in the sense that the
length of its natural parametrization is 2π, which is finite. In contrast the group R
of real numbers is non-compact. For infinitesimal transformations the groups U (1)
and R are essentially identical.
23
|z|=1
1
[23/04/2012]
Paolo Finelli
Corso di Teoria delle Forze Nucleari (2012)
so the application of Φ̂ to a state lowers its nΦ eigenvalue by 1. This is consistent with the
interpretation that the field Φ̂ destroys particles via the â piece or creates an antiparticle via
the b̂† piece. In the same way, considering Φ̂† |NΦ i one easily verifies that Φ† increases nΦ by 1,
creating a particle via ↠or destroying an antiparticle via b̂. The vacuum state (no particles or
antiparticles) is defined by
â(k)|0i = b̂(k)|0i = 0
(2.26)
for all k.
2.1.2
Symmetry breaking potential
If we consider a more general interaction term, the complex Lagrangian can be written as follows
1
L = (∂µ Φ∗ )(∂ µ Φ) − V (Φ) ,
2
(2.27)
1
V ≡ λ(Φ∗ Φ)2 + µ2 (Φ∗ Φ) ,
4
(2.28)
where
with µ2 , λ > 0 (λ must be positive to have a bounded energy spectrum). The Hamiltonian
density is then
H = (∂t Φ∗ )(∂t Φ) + ∇Φ∗ · ∇Φ + V (Φ) .
(2.29)
It is very easy to see that L is invariant under U (1) transformations. As usual, one first consider
the classical case, where the absolute minimum can be reached for: i) Φ = constant and ii)
Φ = Φ0 where Φ0 is the minimum of the classical potential. With the previous choice for µ2 and
λ, the minimum is Φ = 0. In this case we have two degrees of freedom, both massive, and the
vacuum expectation value of the corresponding operator is zero:
h0|Φ̂|0i = 0 ,
(2.30)
because â(k)|0i = b̂(k)|0i = 0. If we change the sign of µ2 , the potential will lead to spontaneous
symmetry breaking (B stands for breaking)
1
V = VB ≡ λ(Φ∗ Φ)2 − µ2 (Φ∗ Φ) .
4
(2.31)
In this case the point Φ1 = Φ2 = 0 is a stationary point, but unstable respect to small fluctuations. The minimum occurs when
(Φ∗ Φ) =
24
2µ2
,
λ
(2.32)
[23/04/2012]
Paolo Finelli
Corso di Teoria delle Forze Nucleari (2012)
or, alternatively
Φ21 + Φ22 =
4µ2
= v2 .
λ
(2.33)
The symmetry breaking condition can be written as
√
|Φ| = v/ 2 .
(2.34)
At this point, it is useful to introduce the polar variables ρ(x) and θ(x), in order to have
ρ(x)
Φ(x) = √ e−iθ(x)/v .
2
(2.35)
The minimum condition is represented by the circle ρ = v: any point on this circle, at any value
of θ, represents a possible classical ground-state (an infinitely degenerate set). If we consider
the excitations about a point on the circle of minima, i.e. ρ = v and θ = 0, we obtain
1
Φ̂(x) = √ (v + ĥ(x))e(−iθ̂(x)/v) ,
2
(2.36)
for the field operator. The Lagrangian becomes3
1
1
µ4
∂µ ĥ∂ µ ĥ − µ2 ĥ2 + ∂µ θ̂∂ µ θ̂ +
+ ...
2
2
λ
3
(2.43)
We have for the kinetic terms
1
1
−i∂µ θ̂ −iθ̂/v
e
∂µ Φ̂ = √ (∂µ ĥ)e−iθ̂/v + √ (v + ĥ)
v
2
2
(2.37)
and
1
i∂µ θ̂ iθ̂/v
1
∂µ Φ̂† = √ (∂µ ĥ)eiθ̂/v + √ (v + ĥ)
e
,
v
2
2
and so the terms which are quadratic in the fields are
∂µ Φ̂† ∂ µ Φ̂ =
1
1
∂µ ĥ∂ µ ĥ + ∂µ θ̂∂ µ θ̂ .
2
2
(2.38)
(2.39)
The potential terms (up to quadratic powers in the field ĥ) are (no θ̂ degrees of freedom)
=
=
µ2
1 1
− λ (v + ĥ)4 +
(v + ĥ)2
4 4
2
λ
µ2 2
µ2 2
− (v 4 + 4v 3 ĥ + 6v 2 ĥ2 + . . .) +
v + µ2 v ĥ +
ĥ
16
2
2
λv 4
λ
3
µ2 2
µ2 2
−
− v 3 ĥ − λv 2 ĥ2 +
v + µ2 v ĥ +
ĥ + . . .
16
4
8
2
2
(2.40)
(2.41)
(2.42)
If we substitute v = 2µ/λ1/2 the linear terms in ĥ cancel.
25
[23/04/2012]
Paolo Finelli
Corso di Teoria delle Forze Nucleari (2012)
The new fields correspond to massive radial oscillations (ρ̂) and angle massless oscillations (θ̂).
The particle spectrum in the spontaneously broken case is very different from that in the normal
case: instead of two degrees of freedom with the same mass µ, one (θ) is massless and the other
√
(h) has a finite mass: 2µ. The broken vacuum |0iB is, of course, annihilated by the operator
âh and âθ . This implies
B h0|Φ̂|0iB
√
= v/ 2 .
(2.44)
This simple model contains the essence of spontaneous symmetry breaking: a non-zero vacuum
value of a field which is not invariant under the symmetry group, zero mass bosons and massive
excitations in a direction of the field space which is orthogonal to the degenerate ground states
(see Fig. 2.1). This model has, of course, a phenomenological origin because the mechanism
V(φ)
Nambu-Goldstone massless boson
Massive scalar boson
§
φ2
φ1
Figure 2.1: Massive and massless excitations in the U (1) model.
µ2 → −µ2 has to be put in by hand. All the vacuum states are good starting point to build the
excited states. If we choose another vacuum: θ = −α, then
B h0, α|Φ̂|0, αiB
v
= e−iα √ = e−iα B h0|Φ̂|0iB .
2
(2.45)
On the other hand
0
e−iα Φ̂ = Φ̂ = Ûα Φ̂Ûα−1 ,
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Corso di Teoria delle Forze Nucleari (2012)
where Ûα = eiαN̂φ . Eq. (2.45) becomes
B h0, α|Φ̂|0, αiB
=B h0|Ûα Φ̂Ûα−1 |0iB ,
(2.47)
and we may interpret Ûα−1 |0iB as the alternative (rotated) vacuum |0, αiB (not in the infinite
volume limit).
For the symmetry current, in terms of ĥ and θ̂,
N̂ µ = v∂ µ θ̂ + 2ĥ∂ µ θ̂ + ĥ2 ∂ µ θ̂/v .
(2.48)
The term involving just the single field θ̂ tells us that there is a non-zero matrix element of the
form
B h0|N̂
µ
|θ, piB = −ipµ ve−ip·x ,
(2.49)
where |θ, pi stands for the state with one θ−quantum state with momentum pµ . When the
symmetry is spontaneously broken, the symmetry current connects the vacuum to a state with
one Goldstone quantum, with an amplitude which is proportional to the symmetry breaking
vacuum expectation value v. ∂µ N̂ µ = 0 only if p2 = 0, as it should be for a Goldstone boson.
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2.2
Higgs mechanism [3]
2.2.1
Local gauge symmetry
Consider the local case: α = α(xµ )4 . If we consider infinitesimal transformations, it is easy to
see that derivative terms change differently from fields:
Φ → Φ + δΦ = Φ − iαΦ
(2.50)
Φ∗ → Φ∗ + δΦ∗ = Φ∗ + iαΦ∗
(2.51)
∂µ Φ → ∂µ Φ + δ(∂µ Φ) = ∂µ Φ − i(∂µ α)Φ − iα(∂µ Φ)
(2.52)
∂µ Φ∗ → ∂µ Φ∗ + δ(∂µ Φ∗ ) = ∂µ Φ∗ + i(∂µ α)Φ∗ + iα(∂µ Φ∗ ) .
(2.53)
If we consider the variation of the free Lagrangian density
L0 = (∂µ Φ∗ )(∂ µ Φ) ,
(2.54)
δL0 = −i(Φ∗ ∂ µ Φ − Φ∂ µ Φ∗ )δ(∂µ α) + total divergence = j µ δ(∂µ α) .
(2.55)
we obtain
The Lagrangian density is not invariant under local U (1) gauge transformation but its variation
depends on the conserved current and the spatial derivatives of the gauge variable α. To mantain
4
te
rac
tio
n
In
tr
me
ym
al s
Loc
The gauge principle, which might also be described as a principle
of local symmetry, is a statement about the invariance properties
Gauge
of physical laws. It requires that every continuous symmetry must
field
be a local symmetry. The key ideas leading up to the introduction of local gauge fields came from Noether, Weyl, and London.
Noether was the first to understand the relation between symmetries and conservation laws. The first attempt to generalize
continuous symmetry for local invariance instead is due to Weyl.
The invariance that Weyl hoped to exploit was an invariance with
respect to change of scale: the requirement that physical laws be
the same if the scale of all length measurements is changed by
Conserved
Symmetry
the same overall factor. Weyl wanted to require a local gauge inquantity
Noether’s Theorem
variance in which the scale changes are allowed to be different at
different points in space and time, analogous to the curvilinear coordinate transformations of general relativity.
In 1927, Fritz London pointed out that the symmetry associated with electric charge conservation was not a scale
invariance, but a phase invariance, i.e. the invariance of quantum theory under an arbitrary change in the complex
phase of the wavefunction. The invariance under a global phase change multiplication of the wavefunction by a
constant phase factor eiθ was trivial in fact; the nontrivial fact was that the existence of the electromagnetic field
allows a much broader kind of invariance, invariance under a local phase change, in which the phase factor varies
arbitrarily from one point to another in space-time. That is, θ becomes an arbitrary function of x, y, z and t.
The word ”gauge” historically refers to a choice of length scale, rather than to the assignment of complex phases.
y
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Corso di Teoria delle Forze Nucleari (2012)
the gauge invariance we must introduce a xµ -dependent term
L1 = −gj µ Aµ ,
(2.56)
where g is a coupling constant and Aµ is the gauge field that transforms like
1
Aµ → Aµ + ∂ µ α .
g
(2.57)
L2 = g 2 Aµ Aµ Φ∗ Φ ,
(2.58)
Finally we have to add
in order to have a locally gauge invariant Lagrange density
δL = δL0 + δL1 + δL2 = 0 .
(2.59)
1
L = (∂µ Φ + igAµ Φ) (∂ µ Φ∗ − igAµ Φ∗ ) − M 2 Φ∗ Φ − Fµν F µν ,
4
(2.60)
Collecting everything we have5
where the term involving Fµν = ∂µ Aν − ∂ν Aµ is clearly gauge invariant and looks like the
Lagrangian density of the Maxwell field. Dµ Φ ≡ ∂µ Φ + igAµ Φ is called covariant derivative
since it transforms under gauge transformations in the same way as the field Φ
δΦ → −iαΦ ,
(2.61)
δ(Dµ Φ) → δ(∂µ Φ) + ig(δAµ )Φ + igAµ (δΦ)
1
= −iα(∂µ Φ) − i(∂µ α)Φ + ig
∂µ α Φ + igAµ (−iαΦ)
g
= −iα(∂µ Φ) + igAµ (−iαΦ) = −iα(∂µ Φ + igAµ Φ) = −iα(Dµ Φ) .
(2.62)
From a geometric point of view we can picture the situation as follows. In order to define the
phase of Φ(x) locally, we have to define a local frame or fiducial field with respect to which the
phase of the field is measured. Local invariance is then the statement that the physical properties
of the system must be independent of the particular choice of frame. In this model the field Φ
can be associated with a particle of charge q = g = e and the conjugate field Φ∗ with a particle
of charge q = −g = −e. The electromagnetic field can therefore be seen as a gauge field that
5
2
a non vanishing mass for the gauge field requires a term of the form MA
Aµ Aµ , but such term is not gauge
invariant, i.e. gauge fields must be massless. For massless fields there are only two degrees of freedom (transverse
d.o.f.).
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Corso di Teoria delle Forze Nucleari (2012)
arises due to the local U (1) symmetry. The equation of motion for the gauge field is
∂ν F νµ = −ig(Φ∗ Dµ Φ − ΦDµ Φ∗ ) ≡ gJ µ ,
(2.63)
and the source term of the electromagnetic field6 is now given by the covariant current J µ rather
than the original current j µ , where ∂µ J µ = 0. We can always promote a global symmetry to
a local (i.e. gauge) symmetry by replacing the derivative operator by the covariant derivative.
Thus, we can make a system invariant under local gauge transformations at the expense of
introducing a vector field Aµ (the gauge field) which plays the role of a connection. From
a physical point of view, this result means that the impossibility of making a comparison at a
distance of the phase of the field Φ(x) requires that a physical gauge field Aµ (x) must be present.
This procedure, which relates the matter and gauge fields through the covariant derivative, is
known as minimal coupling. Let add |Φ|4 self-interaction, in order to have
1
L = (∂µ Φ + igAµ Φ) (∂ µ Φ∗ − igAµ Φ∗ ) − µ2 Φ∗ Φ + λ|Φ∗ Φ|2 − Fµν F µν ,
4
(2.64)
where M 2 → µ2 is now a free parameter. As been done before a new (non trivial) vacuum is
generated (if µ2 < 0)
v
h0|Φ|0i = √
2
with
v=
r
−µ2
.
2λ
(2.65)
Expading around the vacuum, we use polar variables as done before
1
Φ̂(x) = √ (v + ρ̂(x)) e−iθ̂(x)/v .
2
(2.66)
With this parametrization, the covariant derivative reads
1
Dµ Φ̂ = √ e−iθ̂(x)/v ∂µ ρ̂ + ig(ρ̂ + v)B̂µ ,
2
(2.67)
where µ has been replaced by B̂µ = µ + g1 ∂µ α̂ because α̂ = −θ̂(x)/v . The Lagrangian can
be written in the form
1
1
1
L = − F̂µν F̂ µν + MB2 B̂µ B̂ µ + (∂µ ρ̂)2 + µ2 ρ̂2
4
2
2
1 2 2
+ g (ρ̂ + 2ρ̂v)B̂µ B̂ µ − λv ρ̂3 − λρ̂4 ,
2
6
(2.68)
(2.69)
Nonetheless the charge associated to the conservation of electric charge is still given by Qch =
pointed out by authors of Ref. [15].
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Corso di Teoria delle Forze Nucleari (2012)
where now Fµν is expressed in terms of B̂µ -fields. The massless vector field µ and the massless
would-be Goldstone field α̂ have been replaced by a new massive vector field B̂µ (with MB = gv).
√
The mass of the residual neutral Higgs particle, specified by the ρ̂ field, is 2|µ|. The reason
that the would-be Goldstone boson is absorbed in the case of U (1) gauge symmetry is that the
freedom which exists to choose the gauge field Aµ (up to a local phase change α(x)) is exploited
choosing α(x) = −θ(x)/v to eliminate one of the two real scalar fields, α(x) (the other being
ρ(x)). The Goldstone mode θ(x) has been converted into the longitudinal mode of the vector
(gauge) field. One therefore end up with a massive vector boson Bµ (x) (3 degrees of freedom)
plus a massive scalar (neutral) boson ρ(x) (1 degree of freedom): four degrees of freedom as
before (two real scalar fields plus two polarizations for the massless gauge field). To show
explicitly how a Goldstone boson disappears let’s study the equation of motion of the gauge
field:
ν
Âν − ∂ ν (∂µ µ ) = Jˆem
,
(2.70)
ν
Jˆem
= iq(Φ̂† ∂ ν Φ̂ − (∂ ν Φ̂† )Φ̂) − 2q 2 Âν Φ̂† Φ̂ .
(2.71)
where
If we insert Eq. (2.66), we obtain
ν
Jˆem
∂ ν θ̂
 −
vq
2 2
ν
=v q
!
+ higher order terms .
(2.72)
Retaining the linear terms the gauge field satisfies the following equation of motion
ν
ν
µ
2 2
 − ∂ (∂µ  ) = −v q
∂ ν θ̂
 −
vq
ν
!
,
(2.73)
where now a gauge transformation on Âν has the following form
1
0
µ (x) →  µ (x) = µ (x) + ∂µ α̂(x)
q
(2.74)
for arbitray α̂. If we define
0
 µ (x) = µ (x) −
∂ ν θ̂
,
vq
(2.75)
we basically fix the gauge. The resulting equation for µ is
0
0
0
 µ (x) − ∂ ν ∂µ  µ (x) = −v 2 q 2  ν (x)
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Symmetry
Corso di Teoria delle Forze Nucleari (2012)
Mechanism
Original
fields
Physical
fields
Goldstone
1 complex scalar field
(2 d.o.f)
1 massive real scalar
(1 d.o.f.)
1 massless Goldstone mode
(1 d.o.f.)
Higgs
1 complex scalar field
(2 d.o.f)
1 gauge field
(2 d.o.f.)
1 massive real Higgs field
(1 d.o.f.)
1 massive vector field
(3 d.o.f.)
U(1)
that can be interpreted as an equation for a free vector massive field
0
0
( + v 2 q 2 )Â µ (x) − ∂ ν ∂µ Â µ (x) = 0 ,
(2.77)
with mass equal to vq.
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2.3
Corso di Teoria delle Forze Nucleari (2012)
Real scalar fields: SO(3)
We now extend the discussion to a system with a continuous, non-Abelian symmetry such as
SO(3). To that end, we consider the Lagrangian
~ = L(Φ1 , Φ2 , Φ3 , ∂µ Φ1 , ∂µ Φ2 , ∂µ Φ3 )
~ ∂µ Φ)
L(Φ,
1
µ2
λ
=
∂µ Φi ∂ µ Φi − Φi Φi − (Φi Φi )2 ,
2
2
4
(2.78)
where µ2 < 0, λ > 0, with Hermitian fields Φi . The Lagrangian of Eq. (2.78) is invariant under
a global isospin rotation,7
g ∈ SO(3) : Φi → Φ0i = Dij (g)Φj = (e−iαk Tk )ij Φj .
(2.79)
For the Φ0i to also be Hermitian, the Hermitian Tk must be purely imaginary and thus antisymmetric. The iTk provide the basis of a representation of the so(3) Lie algebra and satisfy
the commutation relations [Ti , Tj ] = iijk Tk . We will use the representation with the matrix
elements given by tijk = −iijk . As already done we now look for a minimum of the potential
which does not depend on x and find
~ min | =
|Φ
r
−µ2
≡ v,
λ
~ =
|Φ|
q
Φ21 + Φ22 + Φ23 .
(2.80)
~ min can point in any direction in isospin space we now have a non-countably infinite
Since Φ
number of degenerate vacua. In analogy to the discussion of the last section, any infinitesimal
external perturbation which is not invariant under SO(3) will select a particular direction which,
by an appropriate orientation of the internal coordinate frame, we denote as the 3 direction,
~ min = vê3 .
Φ
(2.81)
7
Of course, the Lagrangian is invariant under the full group O(3) which can be decomposed into its two
components: the proper rotations connected to the identity, SO(3), and the rotation-reflections. For our purposes
it is sufficient to discuss SO(3).
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~ min of Eq. (2.81) is not invariant under the full group G = SO(3) since rotations about
Clearly, Φ
~ min .8 To be specific, if
the 1 and 2 axis change Φ

0


~ min = v 
Φ
 0 ,
1
we obtain

0 0
0
~ min = 
T1 Φ
 0 0 −i
0 i 0

0 0 i

~
T2 Φmin =  0 0 0
~ min
T3 Φ

0


0

 


  0  = v  −i 
v
0
 
 
0
i
 
 
 0  = v 0 
−i 0 0
v
0

 
0 −i 0
0

 
=  i 0 0  0  = 0 .
0 0 0
v
~ min invariant does not form a group,
Note that the set of transformations which do not leave Φ
~ min is invariant under a subgroup
because it does not contain the identity. On the other hand, Φ
F of G, namely, the rotations about the 3 axis:
h∈F :
~ 0 = D(h)Φ
~ = e−iα3 T3 Φ,
~
Φ
~ min = Φ
~ min .
D(h)Φ
(2.82)
We expand Φ3 with respect to v,
Φ3 = v + η,
(2.83)
where η(x) is a new field replacing Φ3 (x), and obtain the new expression for the potential
Ṽ
=
1
λ
λ
(−2µ2 )η 2 + λvη(Φ21 + Φ22 + η 2 ) + (Φ21 + Φ22 + η 2 )2 − v 4 .
2
4
4
(2.84)
8
i.e. T1 and T2 do not annihilate the ground state or, equivalently, finite group elements generated by T1 and
T2 do not leave the ground state invariant.
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Corso di Teoria delle Forze Nucleari (2012)
(x2 + y 2 ) (x2 + y 2 )2
−
V (x, y) =
2
4
Figure 2.2: Two-dimensional rotationally invariant potential
Upon inspection of the terms quadratic in the fields, one finds after spontaneous symmetry
breaking two massless Goldstone bosons and one massive boson:
m2Φ1 = m2Φ2
= 0,
m2η = −2µ2 .
(2.85)
The model-independent feature of the above example is given by the fact that for each of the
two generators T1 and T2 which do not annihilate the ground state one obtains a massless Goldstone boson. By means of a two-dimensional simplification (see the “Mexican hat” potential
shown in Fig. 2.2) the mechanism at hand can easily be visualized. Infinitesimal variations
orthogonal to the circle of the minimum of the potential generate quadratic terms,
i.e. restoring forces linear in the displacement, whereas tangential variations experience restoring forces only of higher orders.
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Paolo Finelli
2.3.1
Corso di Teoria delle Forze Nucleari (2012)
Application of the Goldstone Theorem
~
Given a Hamiltonian operator with a global symmetry group G = SO(3), let Φ(x)
= (Φ1 (x), Φ2 (x), Φ3 (x))
denote a triplet of local Hermitian operators transforming as a vector under G 9 ,
g∈G:
~
~ 0 (x) = ei
Φ(x)
7→ Φ
= e−i
P3
k=1
P3
k=1
αk Q k ~
Φ(x)e−i
P3
l=1
αl Q l
αk Tk ~
~
Φ(x) 6= Φ(x),
(2.86)
where the Qi are the generators of the SO(3) transformations on the Hilbert space satisfying
[Qi , Qj ] = iijk Qk and the Ti = (tijk ) are the matrices of the three dimensional representation
satisfying tijk = −iijk . We assume that one component of the multiplet acquires a non-vanishing
vacuum expectation value:
h0|Φ1 (x)|0i = h0|Φ2 (x)|0i = 0,
h0|Φ3 (x)|0i = v 6= 0.
(2.87)
Then
1. the two generators Q1 and Q2 do not annihilate the ground state
2. to each such generator corresponds a massless Goldstone boson.
In order to prove these two statements let us expand Eq. (2.86) to first order in the αk :
~0 = Φ
~ +i
Φ
3
X
k=1
~ = (1 − i
αk [Qk , Φ]
3
X
~
αk Tk )Φ
k=1
Comparing the terms linear in the αk
i[αk Qk , Φl ] = lkm αk Φm
and noting that all three αk can be chosen independently, we obtain
i[Qk , Φl ] = −klm Φm ,
which, of course, simply expresses the fact that the field operators Φi transform as a vector.
Using klm kln = 2δmn , we find
i
− kln [Qk , Φl ] = δmn Φm = Φn .
2
9
The relation is equivalent to say
[Qi , Φj (x)] = ijk Φk (x) ,
where ijk is a pure totally antisymmetric function of the three indices.
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In particular,
i
Φ3 = − ([Q1 , Φ2 ] − [Q2 , Φ1 ]),
2
(2.88)
with cyclic permutations for the other two cases.
In order to prove that Q1 and Q2 do not annihilate the ground state, let us consider Eq. (2.86)
for α
~ = (0, π/2, 0),

π
~ = 
e−i 2 T2 Φ

cos(π/2)
0
0 sin(π/2)

Φ1


Φ3


 

  Φ2  =  Φ2 
−Φ1
Φ3
cos(π/2)
 

Φ1
Φ3
 

Φ2  =  Φ2 
1
0
− sin(π/2) 0

0 0 1


=  0 1 0 
Φ3
−1 0 0


Φ1
 −i π Q
i π2 Q2 
= e
 Φ2  e 2 2 .

−Φ1
Φ3
From the first row we obtain
π
π
Φ3 = ei 2 Q2 Φ1 e−i 2 Q2 .
Taking the vacuum expectation value
π
π
v = h0|ei 2 Q2 Φ1 e−i 2 Q2 |0i
Q2 |0i =
6 0 is the only possible solution, since otherwise the exponential operator could be replaced
by unity and the right-hand side would vanish. A similar argument shows Q1 |0i =
6 0.
Let us now turn to the existence of Goldstone bosons, taking the vacuum expectation value of
Eq. (2.88)
i
i
0 6= v = h0|Φ3 (0)|0i = − h0| ([Q1 , Φ2 (0)] − [Q2 , Φ1 (0)]) |0i ≡ − (A − B).
2
2
We will first show A = −B. To that end we perform a rotation of the fields as well as the
generators by π/2 about the 3 axis [see Eq. (2.86) with α
~ = (0, 0, π/2)]:

π
~ = 
e−i 2 T3 Φ

−Φ2


Φ1

π
π



Φ1  = ei 2 Q3  Φ2  e−i 2 Q3 ,
Φ3
Φ3
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Corso di Teoria delle Forze Nucleari (2012)
and analogously for the charge operators



−Q2


Q1

π
π



Q1  = ei 2 Q3  Q2  e−i 2 Q3 .
Q3
Q3
We thus obtain
π
π
π
π
3 i 2 Q3
B = h0|[Q2 , Φ1 (0)]|0i = h0| ei 2 Q3 (−Q1 ) |e−i 2 Q{z
e } Φ2 (0)e−i 2 Q3
1
π
i π2 Q3
−i π2 Q3 i π2 Q3
−e
Φ2 (0)e
e
(−Q1 )e−i 2 Q3 |0i
= −h0|[Q1 , Φ2 (0)]|0i = −A,
where we made use of Q3 |0i = 0, i.e., the vacuum is invariant under rotations about the 3 axis.
In other words, the non-vanishing vacuum expectation value v can also be written as
0 6= v = h0|Φ3 (0)|0i = −ih0|[Q1 , Φ2 (0)]|0i
Z
= −i d3 x h0|[J01 (~x, t), Φ2 (0)]|0i.
We insert a complete set of states 1 =
v = −i
Z Z
X
n
R
P
n |nihn|
(2.89)
into the commutator10
d3 x h0|J01 (~x, t)|nihn|Φ2 (0)|0i − h0|Φ2 (0)|nihn|J01 (~x, t)|0i ,
and make use of translational invariance
Z Z
X
= −i
d3 x e−iPn x h0|J01 (0)|nihn|Φ2 (0)|0i − · · ·
= −i
n
Z
X
n
(2π)3 δ 3 (Pn ) e−iEn t h0|J01 (0)|nihn|Φ2 (0)|0i
−eiEn t h0|Φ2 (0)|nihn|J01 (0)|0i .
Integration with respect to the momentum of the inserted intermediate states yields an expression of the form
= −i(2π)3
X
n0
e−iEn t · · · − eiEn t · · · ,
10
R
P
The abbreviation n |nihn| includes an integral over the total momentum p as well as all other quantum
numbers necessary to fully specify the states.
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Corso di Teoria delle Forze Nucleari (2012)
where the n0 indicates that only states with P = 0 need to be considered. Due to the Hermiticity
of the symmetry current operators J µ,a as well as the Φl , we have
cn := h0|J01 (0)|nihn|Φ2 (0)|0i = hn|J01 (0)|0i∗ h0|Φ2 (0)|ni∗ ,
such that
v = −i(2π)3
X
n0
cn e−iEn t − c∗n eiEn t .
(2.90)
From Eq. (2.90) we draw the following conclusions.
1. Due to our assumption of a non-vanishing vacuum expectation value v, there must exist
0 (0)|ni and hn|Φ
states |ni for which both h0|J1(2)
1(2) (0)|0i do not vanish. The vacuum itself
cannot contribute to Eq. (2.90) because h0|Φ1(2) (0)|0i = 0.
2. States with En > 0 contribute (ϕn is the phase of cn )
1
1
cn e−iEn t − c∗n eiEn t =
|cn | eiϕn e−iEn t − e−iϕn eiEn t
i
i
= 2|cn | sin(ϕn − En t)
to the sum. However, v is time-independent and therefore the sum over states with (En >
0, 0) must vanish.
3. The right-hand side of Eq. (2.90) must therefore contain the contribution from states with
zero energy as well as zero momentum, i.e. zero mass. These zero-mass states are the
Goldstone bosons.
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Corso di Teoria delle Forze Nucleari (2012)
Generalization to n-Lie group
Let consider the model in the case of an arbitrary compact Lie group G 11 of order nG resulting
in nG infinitesimal generators. Once again, we start from a Lagrangian
~ ∂µ Φ)
~ = 1 ∂µ Φ
~ · ∂µΦ
~ − V (Φ),
~
L(Φ,
2
(2.95)
~ is a multiplet of scalar (or pseudoscalar) Hermitian fields. The Lagrangian L and thus
where Φ
~ are supposed to be globally invariant under G, where the infinitesimal transformations
also V (Φ)
of the fields are given by
g∈G:
Φi → Φi + δΦi ,
δΦi = −ia taij Φj .
(2.96)
The Hermitian representation matrices T a = (taij ) are again antisymmetric and purely imaginary. We now assume that, by choosing an appropriate form of V , the Lagrangian generates a
spontaneous symmetry breaking resulting in a ground state with a vacuum expectation value
~ min = hΦi
~ which is invariant under a continuous subgroup F of G. We expand V (Φ)
~ with
Φ
11
N −component real scalar field φa (x)(a = 1, . . . , N ).
In this case the symmetry is the group of rotations in N-dimensional space
φa (x) = Rab φb (x)
(2.91)
φa is said to transform like the N −dimensional (vector) representation of the Orthogonal group O(N ). The
elements of the orthogonal group, R ∈ O(N ), satisfy
R1 ∈ O(N ) and R2 ∈ O(N )
→
R1 R2 ∈ O(N )
∃I ∈ O(N ) such that ∀R ∈ O(N )
→
RI = IR = R
∀R ∈ O(N ), ∃R−1 ∈ O(N ) such that
R−1 = RT
(2.92)
where RT is the transpose of the matrix R.
N −component complex scalar field φa (x)(a = 1, . . . , N ).
If the N-component vector φa (x) is a complex field, it transforms under the group of (N × N ) Unitary transformations U
φ0 a(x) = U ab φb (x) .
(2.93)
The complex N × N matrices U are elements of the Unitary group U (N ) and satisfy
U1 ∈ U (N ) and U2 ∈ U (N )
→
U1 U2 ∈ U (N )
∃I ∈ U (N ) such that ∀U ∈ U (N )
→
U I = IU = U
∀U ∈ U (N ), ∃U −1 ∈ U (N ) such that
U −1 = U †
(2.94)
where U † = (U T )∗ .
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~ min , |Φ
~ min | = v, i.e., Φ
~ =Φ
~ min + ~η ,
respect to Φ
2
~
~
~ = V (Φ
~ min ) + ∂V (Φmin ) ηi + 1 ∂ V (Φmin ) ηi ηj + · · · .
V (Φ)
∂Φ
2 ∂Φi ∂Φj
| {z i }
|
{z
}
2
0
m
(2.97)
ij
The matrix M 2 = (m2ij ) must be symmetric and, since one is expanding around a minimum,
positive semidefinite, i.e.,
X
i,j
m2ij ηi ηj ≥ 0 ∀ ~η .
(2.98)
In this case, all eigenvalues of M 2 are non-negative. Making use of the invariance of V under
the symmetry group G,
~ min )
V (Φ
~ min ) = V (Φ
~ min + δ Φ
~ min )
=
V (D(g)Φ
(2.97)
~ min ) + 1 m2ij δΦmin,i δΦmin,j + · · · ,
=
V (Φ
2
(2.99)
one obtains, by comparing coefficients,
m2ij δΦmin,i δΦmin,j = 0.
(2.100)
Differentiating Eq. (2.100) with respect to δΦmin,k and using m2ij = m2ji results in the matrix
equation
~ min = ~0.
M 2δΦ
(2.101)
~ min = −ia T a Φ
~ min , we conclude
Inserting the variations of Eq. (2.96) for arbitrary a , δ Φ
~ min = ~0.
M 2T aΦ
(2.102)
The solutions of Eq. (2.102) can be classified into two categories:
1. T a , a = 1, · · · , nF , is a representation of an element of the Lie algebra belonging to the
subgroup F of G, leaving the selected ground state invariant. In that case one has
~ min = ~0,
T aΦ
a = 1, · · · , nF ,
such that Eq. (2.102) is automatically satisfied without any knowledge of M 2 .
2. T a , a = nF + 1, · · · , nG , is not a representation of an element of the Lie algebra belonging
~ min 6= ~0, and T a Φ
~ min is an eigenvector of M 2
to the subgroup F. In that case T a Φ
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with eigenvalue 0. To each such eigenvector corresponds a massless Goldstone boson.
~ min 6= ~0 are linearly independent, resulting in nG − nH
In particular, the different T a Φ
independent Goldstone bosons12 .
Let us check these results by reconsidering the example of Eq. (2.78). In that case nG = 3 and
nF = 1, generating 2 Goldstone bosons [see Eq. (2.85)].
We conclude this section with a remark. The number of Goldstone bosons is determined by
the structure of the symmetry groups. Let G denote the symmetry group of the Lagrangian,
with nG generators and F the subgroup with nF generators which leaves the ground state after
spontaneous symmetry breaking invariant.
For each generator which does not annihilate the vacuum one obtains a massless
Goldstone boson. The total number of Goldstone bosons equals nG − nH .
12
If they were not linearly independent, there would exist a nontrivial linear combination


nG
nG
X
X
~ min ) = 
~ min ,
~0 =
ca (T a Φ
ca T a  Φ
a=nF +1
a=nF +1
|
{z
:= T
}
such that T is an element of the Lie algebra of H in contradiction to our assumption.
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Bibliography
[1] Symmetries in Physics, Philosophical Reflections, Edited by Katherine Brading and Elena
Castellani, Cambridge Press (2003).
[2] E. Fabri and L.E. Picasso, Quantum Field Theory and Approximate Symmetries, Phys.
Rev. Lett. 16 (1966) 408.
[3] I. J. R. Aitchinson and A. J. G. Hey, Gauge Theories in Particle Physics, Volume II, IOP
(2004).
[4] E.
Witten,
Lectures
delivered
at
the
Institute
for
Advanced
studies
(IAS),
http://www.math.ias.edu/QFT/spring/index.html . J. A. Swieca, Goldstone’s theorem
and related topics, Cargese Lectures in Physics 4 (1970) 215 (available on request from the
lecturer).
These lectures are extremely difficult.
[5] S. Coleman, The Invariance of the Vacuum is the Invariance of the World, Jour. Math.
Phys. 7 (1966) 787.
[6] J. Goldstone, Field Theories with Superconductor Solutions, N. Cim. 19 (1961) 154.
[7] J. Bernstein, Spontaneous Symmetry Breaking, Gauge Theories, the Higgs Mechanism and
All That, Rev. Mod. Phys. 46 (1974) 7.
[8] T. Brauner, Spontaneous Symmetry Breaking and Nambu-Goldstone Bosons in Quantum
Many-Body Systems, Symmetry 2 (2010) 609.
[9] Chuang Liu Classical Spontaneous Symmetry Breaking, Phil. of Sci. 70 (2003) 1219 and
references therein.
[10] S. Scherer and M. R. Schindler, A Chiral Perturbation Theory Primer, [arXiv:hepph/0505265].
43
Paolo Finelli
Corso di Teoria delle Forze Nucleari (2012)
[11] S. Gasiorowicz, Quantum Physics, Wiley Ed.
[12] S. Weinberg, The Quantum Theory Of Fields. Vol. 2: Modern Applications (Cambridge
University Press, Cambridge, 1996).
[13] M. E. Peskin and D. V. Schroeder, An Introduction To Quantum Field Theory, (AddisonWesley, 1995).
[14] David J. Gross, Symmetry in Physics: Wigner’s Legacy, Phys. Tod. 48 (1995) 46.
[15] K. Brading, Which Symmetry? Noether, Weyl, and Conservation of Electric Charge, Studies in History and Philosophy of Science Part B 33 (2002) 3.
More references
[16] a very exhaustive review:
G. S. Guralnik, C. R. Hagen, T. W. B. Kibble, Broken symmetries and the Goldstone
theorem, Adv. in Part. Phys. 2 (1968) 567 (beyond the scope of this lecture).
[17] a high-level book:
F. Strocchi, Symmetry Breaking, Lect. Not. in Phys. 732 (2008) (beyond the scope of this
lecture).
[18] C. Quigg, Spontaneous symmetry breaking as a basis of particle mass, Rep. Prog. Phys. 70
(2007) 1019 (SSB in gauge theories).
44
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Appendix A
History of Spontaneous Symmetry
Breaking
From Ref. [1] and Wikipedia:
Historically, the concept of SSB first emerged in condensed matter physics. The prototype case is
the 1928 Heisenberg theory of the ferromagnet as an infinite array of spin 1/2 magnetic dipoles, with
spin-spin interactions between nearest neighbours such that neighbouring dipoles tend to align. Although
the theory is rotationally invariant, below the critical Curie temperature Tc the actual ground state of the
ferromagnet has the spin all aligned in some particular direction (i.e. a magnetization pointing in that
direction), thus not respecting the rotational symmetry. What happens is that below Tc there exists an
infinitely degenerate set of ground states, in each of which the spins are all aligned in a given direction. A
complete set of quantum states can be built upon each ground state. We thus have many different possible
worlds (sets of solutions to the same equations), each one built on one of the possible orthogonal (in the
infinite volume limit) ground states. To use a famous image by S. Coleman, a little man living inside one
of these possible asymmetric worlds would have a hard time detecting the rotational symmetry of the laws
of nature (all his experiments being under the effect of the background magnetic field). The symmetry
is still there the Hamiltonian being rotationally invariant but hidden to the little man. Besides, there
would be no way for the little man to detect directly that the ground state of his world is part of an
infinitely degenerate multiplet. To go from one ground state of the infinite ferromagnet to another would
require changing the directions of an infinite number of dipoles, an impossible task for the finite little
man. As said, in the infinite volume limit all ground states are separated by a superselection rule.
The same picture can be generalized to quantum field theory (QFT), the ground state becoming the
vacuum state, and the role of the little man being played by ourselves. This means that there may exist
symmetries of the laws of nature which are not manifest to us because the physical world in which we live
is built on a vacuum state which is not invariant under them. In other words, the physical world of our
experience can appear to us very asymmetric, but this does not necessarily mean that this asymmetry
belongs to the fundamental laws of nature. SSB offers a key for understanding (and utilizing) this physical
possiblity.
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The concept of SSB was transferred from condensed matter physics to QFT in the early 1960s, thanks
especially to works by Y. Nambu and G. Jona-Lasinio. Jona-Lasinio. The idea of SSB was introduced
and formalized in particle physics on the grounds of an analogy with the breaking of (electromagnetic)
gauge symmetry in the 1957 theory of superconductivity by J. Bardeen, L. N. Cooper and J. R. Schrieffer
(the so-called BCS theory). The application of SSB to particle physics in the 1960s and successive years
led to profound physical consequences and played a fundamental role in the edification of the current
Standard Model of elementary particles. In particular, let us mention the following main results that
obtain in the case of the spontaneous breaking of a continous internal symmetry in QFT.
Goldstone theorem. In the case of a global continuous symmetry, massless bosons (known as
Goldstone bosons) appear with the spontaneous breakdown of the symmetry according to a theorem first
stated by J. Goldstone in 1960. The presence of these massless bosons, first seen as a serious problem
since no particles of the sort had been observed in the context considered, was in fact the basis for the
solution by means of the so-called Higgs mechanism (see the next point) of another similar problem, that
is the fact that the 1954 Yang-Mills theory of non-Abelian gauge fields predicted unobservable massless
particles, the gauge bosons.
Higgs mechanism. In 1964 three teams proposed related but different approaches to explain how
mass could arise in local gauge theories. These three, now famous, papers were written by Robert Brout
and Franois Englert, Peter Higgs, and Gerald Guralnik, C. Richard Hagen, and Tom Kibble, and are
credited with the prediction of the Higgs boson and Higgs mechanism (or Englert-Brout-Higgs-GuralnikHagen-Kibble mechanism) which provides the means by which gauge bosons can acquire non-zero masses
in the process of spontaneous symmetry breaking. The mechanism is the key element of the electroweak
theory that forms part of the Standard Model of particle physics, and of many models, such as the Grand
Unified Theory, that go beyond it.
Each of these papers is unique and demonstrates different approaches to showing how mass arise in gauge
particles. Over the years, the differences between these papers are no longer widely understood, due to the
passage of time and acceptance of end-results by the particle physics community. While first to publish
by a couple months, Higgs, Brout and Englert solved half of the problem massifying the gauge particle.
Guralnik, Hagen and Kibble, while published a couple months later, had a more complete solution.
Additional bibliography:
SSB1 L. N. Cooper, Bound Electron Pairs in a Degenerate Fermi Gas, Phys. Rev. 104 (1956)
1189.
SSB2 J. Bardeen, L.N. Cooper and J. R. Schrieffer, Microscopic Theory of Superconductivity,
Phys. Rev. 106 (1957) 162.
SSB3 J. Bardeen, L. N. Cooper and J. R. Schrieffer, Theory of Superconductivity, Phys. Rev.
108 (1957) 1175.
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SSB4 Y. Nambu and G. Jona-Lasinio, Dynamical Model of Elementary Particles Based on an
Analogy with Superconductivity. I, Phys. Rev. 122 (1961) 345.
SSB5 Y. Nambu and G. Jona-Lasinio, Dynamical Model of Elementary Particles Based on an
Analogy with Superconductivity. II, Phys. Rev. 124 (1961) 246.
SSB6 J. Goldstone, A. Salam and S. Weinberg, Broken Symmetries, Phys. Rev. 127 (1962) 965.
SSB7 P. W. Anderson, Plasmons, Gauge Invariance, and Mass, Phys. Rev. 130 (1963) 439.
SSB8 A. Klein and B. W. Lee, Does Spontaneous Breakdown of Symmetry Imply Zero-Mass
Particles?, Phys. Rev. Lett. 12 (1964) 266.
SSB9 W. Gilbert, Broken Symmetries and Massless Particles, Phys. Rev. Lett. 12 (1964) 713.
SSB10 F. Englert and R. Brout, Broken Symmetry and the Mass of Gauge Vector Mesons, Phys.
Rev. Lett. 13 (1964) 321.
SSB11 P. W. Higgs, Broken Symmetries and the Masses of Gauge Bosons, Phys. Rev. Lett. 13
(1964) 508.
SSB12 G. S. Guralnik, C.R. Hagen and T.W.B. Kibble, Global Conservation Laws and Massless
Particles, Phys. Rev. Lett. 13 (1964) 585.
SSB13 P. W. Higgs, Broken Symmetries, Massless Particles and Gauge Fields, Phys. Lett. 12
(1964) 132.
SSB14 P. W. Higgs, Spontaneous Symmetry Breakdown without Massless Bosons, Phys. Rev.
145 (1966) 1156.
SSB15 S. Weinberg, Conceptual Foundations of the Unified Theory of Weak and Electromagnetic
Interactions, Nobel Prize Lecture (1979).
SSB16 P. W. Higgs, My Life as a Boson,
www.kcl.ac.uk/nms/depts/physics/news/events/MyLifeasaBoson.pdf .
SSB17 G. S. Guralnik, The History of the Guralnik, Hagen and Kibble development of the Theory
of Spontaneous Symmetry Breaking and Gauge Particles, [arXiv:0907.3466].
As pedagogical introductions we suggest the following links
1. http://www.vega.org.uk/video/programme/76
2. http://www.hep.ucl.ac.uk/ djm/higgsa.html
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Appendix B
Group Theory - a short introduction
A group G is a set of elements {g, h, k, . . .} for which a multiplication is defined which assigns to
every two elements g, h ∈ G an element g · h which is again an element of the group. In addition
the following properties should hold:
1. The multiplication is associative, which means that we have (g · h) · k = g · (h · k) for all
g, h, k ∈ G. In the special case that the group multiplication is commutative, g · h = h · g
for all g, h ∈ G, the group is called abelian.
2. The set of elements of G contains the identity I, for which we have I · g = g · I for all
g ∈ G, as well as the inverse elements g −1 for every g ∈ G, i.e. g −1 · g = g · g −1 = I.
A subset H of elements contained in G is called a subgroup of G if H itself is also a group
according to the definition given above. Symmetry transformations always form a group. One
can make an obvious distinction between discrete and continuous transformations. Discrete
symmetries usually constitute a finite group, i.e. a group consisting of a finite number of elements. Continuous symmetries depend on one or more parameters in a continuous fashion.
Clearly a group of such transformations contains an infinite number of elements. The dimension
of a continuous group is defined as the number of independent parameters on which the group
elements depend. If the dependence on these parameters is analytic then we are dealing with a
so called Lie group.
If there is a mapping from a group G to a set of matrices D(G) which preserves the group
multiplication then D(G) is called a representation of the group G. In that case, to any element
g ∈ G there belongs a matrix D(g) ∈ D(G) such that to the product g · h of two elements g and
h of G there belongs a matrix D(g · h) such that
D(g · h) = (D(g) · D(h)) ∈ D(G).
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The mapping between G and D(G) is called a homomorphism. If the mapping is one-to-one
then it is called an isomorphism and D(G) is a faithful representation. The number of such
matrices is equivalent to the rank of the group and it is independent of the dimension N of
the representation that will be chosen. In case the mapping is not into a set of matrices but
into some other algebraic structure, it is called a realization. In general a group can have many
different representations.
Example. As an example we recall the representations of the rotation group, which are wellknown from quantum mechanics. These representations are characterized by an integer l(l =
0, 1, 2, . . .) and consist of (2l+1)×(2l+1) matrices acting on states with total angular momentum
L2 = ~2 l(l + 1); the latter are labelled by their value of angular momentum projected along a
certain axis (e.g. Lz = −~l, −~(l − 1), . . . , ~l). For each rotation g (which is a 3 × 3 orthogonal
matrix) there is a (2l + 1) × (2l + 1) matrix D(g), which specifies how the 2l + 1 states transform
among themselves as a result of the rotation. The quantity 2l + 1 is called the dimension of the
representation. It is rather obvious that combining two representations of dimension 2l1 + 1 and
2l2 + 1 leads to another representation of dimension 2(l1 + l2 + 1). The latter representations
are called reducible as they can be reduced to smaller representations. Evidently not much new
is to be learnt from studying reducible representations, so that one usually restricts oneself to
irreducible representations. It can be shown that finite groups must have a finite number of
irreducible representations. Continuous groups have infinitely many representations.
B.1
Lie Groups
Consider a one-parameter Lie group with elements g(ξ). Because of the analyticity of g(ξ) it is
always possible to choose a so-called canonical parametrization, which satisfies
g(ξ 1 )g(ξ 2 ) = g(ξ 1 + ξ 2 ) .
(B.1)
Consequently
g(0) = I ,
g −1 (ξ) = g(−ξ) .
(B.2)
Using this parametrization and the fact that g(ξ) is analytic we can write an element in a
neighbourhood of the identity element as
g(ξ) = I + ξt + O(ξ 2 ) ,
(B.3)
where t is an operator which generates the infinitesimal group transformation. Using (B.1) we
can formally construct finite elements g(ξ) by making an infinite series of infinitesimally small
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steps away from the identity element:
n
g(ξ) = {g(ξ/n)} = lim
n→∞
n
ξ
I + t...
= exp(ξt) ,
n
(B.4)
where the exponentiation is defined by its series expansion. The result can directly be extended
to an n-parameter Lie group:
g(ξ 1 , ξ 2 , . . . ξ n ) = exp(ξ a ta ) ,
(B.5)
where we have adopted the summation convention The quantities ta , which characterize the
infinitesimal transformations that are linearly independent, are called the generators of the Lie
group.
Example. As a second example consider all two-dimensional rotations, which obviously form
a Lie group with the angle of rotation ξ as a natural canonical parameter. Using polar coordinates with x = r cos θ , y = r sin θ a (clockwise) rotation g(ξ) changes the value into θ − ξ.
Infinitesimally one has
x
y
!
→
x
y
!
+ξ
y
−x
!
+ O(ξ 2 ) .
(B.6)
According to the previous relation the generator t can be written as a 2 × 2 matrix
0
1
−1 0
!
.
(B.7)
Using t2 = −I a finite rotation g(ξ) can be written as
∞
X
1 n n
ξ t
g(ξ) = exp(ξt) =
n!
n=0
∞ X
1
1
2 n
n 2n+1
=
(−ξ ) I +
(−1) ξ
t
(2n)!
(2n + 1)!
n=0
!
cos ξ sin ξ
= cos ξI + sin ξt =
,
− sin ξ cos ξ
(B.8)
which indeed constitutes a general two-dimensional rotation. The above group is called SO(2).
It is the group of all orthogonal 2 × 2 matrices with unit determinant. This is a special case of
the group O(N ) which consists of all orthogonal N × N matrices, and the group SO(N ) for the
subgroup of elements of O(N ) with unit determinant. Similarly, U (N ) is the group of unitary
N ×N matrices, and SU (N ) is the group of elements of U (N ) with unit determinant. Obviously,
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O(N ) and SO(N ) are subgroups of U (N ) and SU (N ), respectively.
B.2
Lie Algebra
Consider an n-parameter Lie group G with elements g(ξ 1 , . . . ξ n ) and generators ta (a = 1, . . . , n).
According to (B.5) we may write
g(ξ 1 , . . . ξ n ) = exp(ξ a ta ) .
(B.9)
A product of two such elements can be expressed by means of the Baker-Campbell-Hausdorff
formula
g(ξ 1 , . . . ξ n ) · g(χ1 , . . . χn ) = exp(ξ a ta ) · exp(χb tb )
1
1 a b c
= exp{ξ a ta + χa ta + ξ a χb [ta , tb ] +
ξ ξ χ + χa χb ξ c [ta , [tb , tc ]]
2
12
+ higher order commutators of t} .
(B.10)
Because G is a group, the product (B.10) must again be an exponential form of the generators,
so there must be coefficients η 1 , . . . , η n such that
g(ξ 1 , . . . , ξ n ) · g(χ1 , . . . χn ) = exp(η a ta ) .
(B.11)
This is possible if and only if any commutator of generators can again be written as a linear
combination of generators. In other words, the generators must close under commutation:
c
[ta , tb ] = fab
tc ,
(B.12)
c are constants, antisymmetric in their lower indices (since we assume real parameters
where fab
ξ a , these constants are real). With this property the generators ta form the basis of the so-called
Lie algebra g associated with the Lie group G. Therefore they are called the structure constants
of the group (abelian groups have zero structure constants).
Example. As an example consider the group SU (2), defined as the set of all unitary 2 × 2
matrices with unit determinant. Elements of this group can be written as (for SU (2) we prefer
to include a factor i in order to have hemitean instead of antihermitean generators)
~ = exp(iξa ta ) .
gSU (2) (ξ)
(B.13)
~ is a unitary matrix with unit determinant leads to the following
The requirement that gSU (2) (ξ)
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conditions for the generators:
1
ta = τa .
2
(B.14)
τi τj = δij 1 + iijk τk
(B.15)
These τ matrices obey
and are hermitean (τi = τi† ) and traceless (Trτi = 0). Since [τi , τj ] = i2ijk τk is easy to prove
that
[ta , tb ] = iabc tc .
(B.16)
For infinitesimal transformations
~ ' 1 + iB + O(B 2 )
g(ξ)
B = ξi
τi
2
(B.17)
where B † = B and TrB = 0. The finite transformation is found by exponentiation of Eq. (B.17):
n
τ ξ
τ
i
i
i
~ = lim 1 + i
g(ξ)
.
= exp iξi
n→∞
n 2
2
The matrices 12 τi are the generators of the rotations for the l =
~ =
g(ξ)
∞
X
n=0
1
(2n)!
iξj τj
2
2n
+
∞
X
n=0
1
(2n + 1)!
1
2
(B.18)
representation. By definition
iξj τj
2
2n+1
.
(B.19)
With the help of
(iξj τj )2 = ξj ξk τj τk = −ξ 2 1 ,
(B.20)
one can show that
(iξj τj )2n = (−)n ξ 2n 1
and
(iξj τj )2n+1 = (−)n ξ 2n (iξj τj ) .
(B.21)
With the previous relations we can define the SU (2) transformation element as
~ = cos ξ 1 + i sin ξ ξj τj ,
g(ξ)
2
2 ξ
where ξ =
p
(B.22)
ξ12 + ξ22 + ξ32 . Every 2 × 2 matrix can be decomposed in th unit matrix 1 and τi :
g = c0 1 + ici τi ,
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1
c0 = Tr(g)
2
In our case the coefficients are
c0 = cos
1
and ci = Tr(gτi ) .
2
(B.24)
ξ
2
(B.25)
ci =
ξi
ξ
sin
ξ
2
with the properties c20 + c2i = 1. Eq. (B.23) can also be written in terms of two complex
parameters a and b
g=
a
b
−b∗ a∗
!
,
(B.26)
with |a|2 + |b|2 = 1. Such matrices of this form form elements of the group SU (2), the group of
unitary 2 × 2 matrices with determinant 1 because they obey
g † = g −1
detg = 1 .
(B.27)
The parameter space parametrized by ξ can be restricted to the inside of a 2-dimensional sphere
of radius 2π, i.e.
(ξ 1 )2 + (ξ 2 )2 + (ξ 3 )2 ≤ (2π)2 .
(B.28)
The parameter space of SU(2) can be divided into two parts: an inside region where ξ ≤ π and
an outer shell with π < ξ ≤ 2π. To each point ξ in the first region one can assign a point ξ 0 in
the second region, such that both are located on a straight line passing through the origin and
separated by a distance 2π (so that they are in opposite directions); explicitly
2π − ξ ~
ξ 0 ≤ ξ ≤ π , π ≤ ξ 0 ≤ 2π .
ξ~0 = −
ξ
(B.29)
The SU (2) elements corresponding to ξ~ and ξ~0 are then related by
~ .
gSU (2) (ξ~0 ) = −gSU (2) (ξ)
B.3
(B.30)
Representations
Suppose that one can find n matrices Ya with the same commutation relations as the elements
of some Lie algebra g:
c
[Ya , Yb ] = fab
Yc
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ξ2
ξ3 =0
■
●
▲
g=1
▲
■
●
π
2π
ξ1
g=-1
Figure B.1: Parameter space with ξ3 = 0. The origin corresponds to the identity 1. The
boundary is a circle with radius 2π which corresponds to the SU (2) element g = −1. Each
element inside the inner circle, which has radius π has a corresponding element in the outer region
which differs by an overall minus sign. The curve connecting the two filled circles corresponds
to a continuous set of SU (2) transformations differing by an overall sign. The parameter space
of SO(3) can be imbedded in the same plot and covers only the inside of the circle with radius
π. Two opposite points on the inner circle corresponds to the identical element of SO(3). The
SO(3) elements corresponding to the solid curve describe a closed continuous set of SO(3)
transformations.
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then by definition, these matrices form the basis of a representation of this Lie algebra. Exponentiation of linear combinations of Ya leads to a representation of the corresponding Lie group
G:
g(ξ 1 , . . . , ξ n ) → exp(ξ a Ya ) ,
(B.32)
c completely determine the Lie group. Thus, each representation of the Lie
because the fab
algebra induces a representation of the corresponding group. For each Lie group there is a
special representation called the adjoint representation, which has the same dimension as
the group itself. This follows from the Jacobi identity which holds for any three matrices A, B
and C:
[[A, B], C] + [[B, C], A] + [[C, A], B] = 0 .
(B.33)
Choosing A = ta , B = tb and C = tc , one obtains the Jacobi identity for the structure constants
e d
e d
e d
fea + fca
feb = 0 .
fec + fbc
fab
(B.34)
If now regard the structure constants as elements of n × n matrices fa according to
c
(fa )cb ≡ fab
,
(B.35)
we can rewrite Eq. (B.34) as a matrix identity
−(fc )de (fa )eb + (fa )de (fc )eb − (fa )ec (fe )db = 0
(B.36)
or, after relabeling of indices,
c
([fa , fb ])de = fab
(fc )de .
(B.37)
Consequently the matrices fa generate a representation of the Lie algebra and therefore of the Lie
group; clearly, the adjoint representation has dimension n, like the Lie group itself. Obviously an
abelian group has vanishing structure constants, so that its adjoint representation is trivial, i.e.
it consists of the identity element. As an example consider again SU (2). As shown above this
c equal to i
group has three generators ta and structure constants fab
abc . Therefore the adjoint
representation is 3-dimensional with generators Sa , given by
(Sa )cb = iabc
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or, explicitly

0 0
0



S1 =  0 0 −i  ,
0 i 0

0
0 i


0 0  ,
−i 0 0

S2 =  0

0 −i 0

S3 =  i
0
0
0


0  .
0
(B.39)
which are just the generators of SO(3). Hence SU (2) and SO(3) have the same structure
constants and are therefore locally equivalent. This fact is of physical importance because
rotations of spatial coordinates are governed by SO(3), while spin rotations are described by
SU (2); hence spatial and spin rotations form different representations of the same group. It is
interesting to compare the parameter space of SO(3) to that of SU (2) The parameter space of
SU(2) covers the group SO(3) twice, because two points ξ and ξ 0 satisfying Eq. B.29 correspond
to the same element of SO(3). For this reason the parameter space of SO(3) can be restricted
to the region with ξ ≤ π. Opposite points on the 2-dimensional sphere with radius π that forms
the boundary of this region correspond to the same SO(3) element. The corresponding SU (2)
elements differ by a sign (B.30). Finite transformations in the adjoint representation can be
obtained by exponentiation. Hence one has n × n transformation matrices defined by
gadj (ξ) = exp(ξ a fa ) .
(B.40)
Quantities transforming in this representation are n−dimensional vectors φa . A convenient way
of dealing with such vectors is based on a matrix notation
Φ = φa ta ,
(B.41)
where ta are the group generators in some arbitrary representation. The transformation
Φ → Φ0 = gΦg −1 ,
(B.42)
with g in the same representation as ta (so that g = exp(ξ a ta )), now induces the same transformation on the φa , i.e.
g(φa ta )g −1 = φ0a ta ,
(B.43)
φ0a = (gadj φ)a .
(B.44)
with
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Spin 0
(scalar)
SO(3)
Symmetry group: rotations
SU(2)
Covering group
Spin 1/2
(spinor)
Spin 1
(vector)
...
...
Irreducible representations
Types of particles
Figure B.2: Because quantum mechanical states are rays in a Hilbert space, Wigner was led
from the group of rotations SO(3) to its covering group SU (2). The irreducible representations
of SU (2) then correspond to types of particles.
Casimir operator
The Casimir operator is a quantity which is invariant in any representation, i.e.
gCg −1 = C ,
(B.45)
where g denotes any group element in the representation corresponding to ta . For the rotation
group SO(3) the Casimir operator is just the total angular momentum operator (modulo factor
2~2 ), and one has
1
C = l(l + 1)I
2
(B.46)
where l labels the representations.
Bibliography.
1. H. Georgi, Lie Algebras in Particle Physics, (Benjamin/Cummings, Reading, MA, 1982).
2. G. ‘t Hooft, Lie Groups in Physics,
http://www.phys.uu.nl/~thooft/lectures/lieg07.pdf
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Appendix C
Wigner theorem
From Symmetry in Physics: Wigner’s Legacy by D.J. Gross [14].
Wigner started from the description of quantum mechanical states as unit rays1 in a Hilbert
space: [eiθ ]ψ, where |ψ|2 = 1 and θ ranges from 0 to 2π. Noting that physical predictions are
given by transition probabilities, |hψ, φi|, he defined a symmetry transformation as a map
preserving |hψ, φi|, i.e.
T : ψ → ψ0 = T ψ ,
(C.1)
|hψ|φi|2
|hψ 0 |φ0 i|2
=
hψ 0 |ψ 0 ihφ0 |φ0 i
hψ|ψihφ|φi
(C.2)
This does not completely define the operator T since we can always rescale T by a complex
number Zφ and still obtain the same physical state, i. e. T |φi and Zφ T |φi represent the same
physical state. What Wigner showed was that if U is an operator satisfying
|hT ψ|T φi|2
|hψ|φi|2
=
,
hT ψ|T ψihT φ|T φi
hψ|ψihφ|φi
(C.3)
then one can always adjust the phases so that T is either a unitary operator or an anti-unitary
operator. A unitary operator T is such that
hT ψ|T φi = hψ|T † T |φi = hψ|φi
1
(C.4)
Consider a quantum theory formulated on a Hilbert space H. A physical state corresponds to a ray R in the
Hilbert space, where a ray is defined as a set of normalized vectors (hψ|ψi = 1), where |ψi and |ψ 0 i belong to the
same ray if they are equal up to a phase (i.e., if |ψ 0 i = eiθ |ψi for some real θ). The notation |ψi ∈ R indicates
that |ψi belongs to the ray R and we define R(ψ) to denote the ray that contains the vector |ψi. We will consider
a transformation T defined on physical states, so T maps one ray onto another. The abbreviation T (ψ) denotes
T (R(ψ)), the image under T of the ray that contains the vector |ψi.
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for all states |φi, |ψi, so that
T †T = T T † = 1
(C.5)
T (α|φi + β|ψi) = αT |φi + βT |ψi
(C.6)
and is linear so that
where α, β are arbitrary complex numbers. In that case,
|hψ|T † T φi|2
|hT ψ|T φi|2
|hψ|φi|2
=
=
hT ψ|T ψihT φ|T φi
hψ|ψihφ|φi
hψ|T † T ψihφ|T † T φi
(C.7)
as required. An anti-unitary operator is such that
hT ψ|T φi = hψ|φi∗ = hφ|ψi
(C.8)
T (α|φi + β|ψi) = α∗ T |φi + β ∗ T |ψi
(C.9)
and is anti-linear so that
for arbitrary complex numbers α, β. In that case,
|hφ|ψi|2
|hψ|φi|2
|hT ψ|T φi|2
=
=
hT ψ|T ψihT φ|T φi
hψ|ψihφ|φi
hψ|ψihφ|φi
(C.10)
as required. The second possibility is required for the description of time-reversal invariance, in
which the symmetry transformation interchanges initial and final states. In classical mechanics,
symmetries of the equations of motion can be used to derive new solutions. Thus if the laws of
motion are invariant under spatial rotations and x(t) is a solution of the equations of motion, say
an orbit of the Earth around the Sun, then Rx(t), the spatially rotated x(t), is also a solution.
Wigner understood that in quantum theory, invariance principles permit even further reaching
conclusions than in classical mechanics. In quantum mechanics there is a new and powerful
feature due to the linearity of the symmetry transformation and the superposition principle.
Thus if |Ψi is an allowed state then so is R|Ψi, where R is the operator in the Hilbert space
corresponding to the symmetry transformation R. So far this is similar to classical mechanics.
However, we can now superpose these states, that is, construct a new allowed state: |Ψi + R|Ψi.
There is no classical analog for such a superposition of, say, two orbits of the earth. As Wigner
pointed out, the superposition principle means that we can construct linear combinations of
states that transform in a simple way under the symmetry transformations. Thus superimposing
P
all states that are related by rotations, we obtain a state |Φi = R R|Ψi that is rotationally
invariant:
R|Φi =
X
R0
RR0 |Ψi =
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X
R00
R00 |Ψi = |Φi .
(C.11)
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The state |Φi forms a singlet representation of the rotation group. Other superpositions of
rotated states will yield other irreducible representations of the symmetry group. lrreducible
representations are special: they cannot be further subdivided, any subset of states gets mixed
by the symmetry group with all the other states of the representation. Furthermore any state
can be written as a sum of states transforming according to irreducible representations of the
symmetry group. Wigner realized that these special states can be used to classify all the states
of a system possessing symmetries, and play a fundamental role in the analysis of such systems.
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Appendix D
Noether theorem
The existence of continuous symmetries implies the existence of conservation laws. One important consequence is the existence of locally conserved currents.
Noether’s theorem: For every continuous global symmetry there exists a global
conservation law.
Before showing the proof of Noether theorem, let’s discuss the connection that exists between
locally conserved currents and constants of motion. Let J µ (x) be some locally conserved current,
i.e.
∂µ J µ (x) = 0 .
(D.1)
Let Ω be a bounded 4−volume of space-time with boundary ∂Ω. The Gauss Theorem tells us
that
0=
Z
4
µ
d x ∂µ J (x) =
Ω
I
dSµ J µ (x) ,
(D.2)
∂Ω
where the right hand side is a surface integral on the oriented closed surface ∂Ω (a 3−volume).
Let Ω be a 4−volume which extends all the way to infinity in space but which has a finite extent
in time ∆T . If there are no currents in the large volume limit (lim|x|→∞ J µ (x, x0 ) = 0), then
only the top and the bottom of the boundary ∂Ω contribute to the surface integral. Hence
0=
Z
V (T +∆T )
3
0
d x J (x, T + ∆T ) −
Z
d3 x J 0 (x, T ) ,
(D.3)
V (T )
where we assumed dS0 ≡ d3 x. Thus we see the quantity Q(T )
Q(T ) =
Z
d3 x J 0 (x, T ) ;
V (T )
is a constant of motion, i.e. Q(T + ∆T ) = Q(T ) for every ∆T .
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V(T+ΔT)
∂Ω
Time
T+ΔT
(surface)
Ω
T
(volume)
V(T)
Space (R3)
The existence of a locally conserved current
∂µ J µ (x) = 0
(D.5)
implies the existence of a conserved charge Q
Q=
Z
d3 x J 0 (x, T ) ,
(D.6)
which is a constant of motion.
The proof of Noether’s theorem reduces to the proof of the existence of a locally conserved
current. We proceed with a simple proof using complex scalar field φ(x) 6= φ∗ (x) and considering
internal symmetry. The system has the continuous global symmetry
φ(x) → φ0 (x) = eiα φ(x)
∗
0∗
−iα ∗
φ (x) → φ (x) = e
φ (x) ,
(D.7)
(D.8)
where α is a real number. The system is invariant if L satisfies
L(φ0 , ∂µ φ0 ) ≡ L(φ, ∂µ φ) .
(D.9)
In particular for infinitesimal transformations we have
φ0 (x) = φ(x) + δφ = φ(x) + iαφ(x)
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φ0∗ (x) = φ∗ (x) + δφ∗ = φ(x) − iαφ(x) .
(D.11)
If L is invariant, then variation δL
δL =
δL
δL
δL
δL
+
+ ∗+
δφ δ∂µ φ δφ
δ∂µ φ∗
(D.12)
must be zero. Using the equations of motion
δL
− ∂µ
δφ
δL
δ∂µ φ
=0
(D.13)
and its complex conjugate, the variation of L can be written in the form of a total divergence
δL
δL ∗
δL = ∂µ i
φ−
φ α .
δ∂µ φ
δ∂µ φ∗
(D.14)
Since α is arbitrary, δL = 0 if and only if the current
µ
J =i
δL
δL ∗
φ−
φ
δ∂µ φ
δ∂µ φ∗
(D.15)
is conserved, i.e. ∂µ J µ (x) = 0.
Let’s discuss some consequences of the Noether’s theorem in the quantum context. We consider
linear Hermitian operators in quantum mechanics. Each operators could play a double role: on
one hand they can represent dynamical variables of the theory1
hψ|Â|ψi = hAi
(D.16)
and on other hand they can serve as generators of a class of transformations
|ψi → |ψ 0 i = e−iλ |ψi ,
(D.17)
where λ is a real parameter. Hermitian linear operators generally do not commute with each
other and whether or not two operators commute has deep physical significance.
If  and B̂ commute then
1. The transformations generated by  and B̂ commute with each other (i.e. translations
commute, rotations do not).
1
 is the operator, A is the dynamical variable and |ψi is vector state.
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2. The dynamical variable A is invariant under the transformations generated by B̂:
hψ 0 |Â|ψ 0 i = hψ|Â|ψi = hAi
|ψ 0 i = e−iλB̂ |ψi .
(D.18)
The same is true if we exchange A and B.
3. The dynamical variables A and B are simultaneously measurable with arbitrary precision, i.e. there is no uncertainty principle relating them (they have a complete set of
simultaneous eigentates).
Let’s suppose that B̂ = Ĥ, the Hamiltonian operator. The transformations generated by Ĥ are
time displacements
i~
d
|ψi = Ĥ|ψi
dt
|ψ(t)i = e−iĤt |ψ(0)i .
→
(D.19)
In this case we can make the following statements:
1. If the dynamical variable A is invariant under the transformations generated by Ĥ is
equivalent to say that A is a constant of motion, i.e. its expectation value in any state will
be invariant under any time displacement.
2. If H is invariant under transformations generated by  is equivalent to say that A represents a symmetry of dynamical laws.
3. The dynamical variable A is conserved if and only if the dynamical laws (generated by Ĥ)
are invariant under the transformations generated by Â.
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