Download Invitation to Local Quantum Physics

Invitation to Local Quantum Physics
Jakob Yngvason
University of Vienna
Madrid, December 2–3, 2013
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
1 / 73
Topics
Relativistic symmetries in quantum theory
Problems with position operators
The framework(s) of local quantum physics
The structure of local algebras
Entanglement and causal independence
From local algebras to scattering of particles
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
2 / 73
Relativistic symmetries in quantum theory
Symmetry group of (special) relativistic physics:
Inhomogeneous Lorentz group, generated by translations a of
Minkowski space M = R4 and Lorentz transformations Λ with
(a1 , Λ1 ) ◦ (a2 , Λ2 ) = (a1 + Λ1 a2 , Λ1 Λ2 ).
Excluding reflections for the moment we focus on the proper,
orthochronous Lorentz group L↑+ ; its inhomogeneous version is the
↑
proper, orthochronous Poincaré group, P+
.
According to E. Wigner (1939) this group should, in every relativistic
quantum theory, have a unitary representation, possibly up to a phase,
on the Hilbert space of state vectors.
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
3 / 73
The phase can be eliminated by replacing L↑+ by its universal covering
group SL(2, C). This is the group of complex 2 × 2 matrices of
determinant 1.
Recall its operation on vectors x = (x0 , . . . , x3 ) in Minkowski space:
Define the hermitian matrix x = σ µ xµ where the σ µ are Pauli matrices.
e
Then
x 7→ x0 = AxA∗
e
e
e
defines a transformation Λ(A) in L↑+ for every A ∈ SL(2, C). This gives
rise to a double covering of L↑+ because Λ(A) = Λ(B) if and only
A = ±B.
The group product in the inhomogeneous SL(2, C) (ISL(2, C)) is
(a, A) ◦ (b, B) = (a + Λ(A)b, AB).
A classification of its representations was achieved by E. Wigner in a
ground-breaking paper in 1939.
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
4 / 73
Energy and momentum
Translations of Minkowski space, i.e., the transformations (a, 1), form a
↑
commutative subgroup of P+
that in every unitary representation U
has self-adjoint infinitesimal generators P µ :
U (a, 1) = eiaµ P
µ
The generator of time translations, P 0 , is the energy operator, i.e., the
Hamiltonian, the other three, P = (P 1 , P 2 , P 3 ), are the components of
the momentum operator.
spec U (a, 1), i.e., the joint spectrum of the P µ , is a Lorentz invariant
subset of the (dual) Minkowski space.
The spectrum condition is the stability requirement that P 0 is
bounded below; this is equivalent to the condition that the joint
spectrum of the P µ ’s is contained in the forward light cone V+ .
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
5 / 73
The joint diagonalization of the P µ leads to a integral decomposition of
the Hilbert space of state vectors:
Z ⊕
H=
Hp dµ(p)
where dµ(p) is a Lorentz invariant measure with support in the forward
light cone.
Its most general form is
Z
0
µ
2
2
dµ(p) = cδ(p) + θ(p )δ(pµ p − m )dρ(m ) dp
The first term corresponds to vacuum state(s) with zero energy and
momentum. A contribution for a fixed m2 corresponds to states of
mass m ≥ 0 as an eigenvalue of the Mass Operator M = (Pµ P µ )1/2 .
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
6 / 73
Internal degrees of freedom
Focus now on a fixed m. All the spaces Hp with p on the same mass
hyperboloid
+
Hm
= {p : pµ pµ = m2 , p0 ≥ 0}
can be identified with each other and written as
Hint ⊗ Htrans
with
Htrans = L2 (R4 , dµm (p);
dµm (p) = θ(p0 )δ(pµ pµ − m2 ).
+ →H
Equivalently, the space consists of “wave functions” ψ : Hm
int
↑
that are square integrable w.r.t. dµm (p). The action of P+ be written as
µ
(U (a, Λ)ψ)(p) = eiaµ p W (p, Λ)ψ(Λ−1 p)
where the unitary operators W (p, Λ) on Hint satisfy the “cocycle
relation”
W (p, Λ1 )W (Λ−1
1 p, Λ2 ) = W (p, Λ1 Λ2 ).
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
7 / 73
Stabilizer groups (“little groups”)
Note that if Λ−1
1 p = p, then W (p, Λ1 )W (p, Λ2 ) = W (p, Λ1 Λ2 ).
↑
The further analysis of the possible representations of P+
now
proceeds by considering the stabilizer groups (“little groups”) for
+ . These are the subgroups of SL(2, C)
some standard vectors p̄ ∈ Hm
that leave invariant the vectors p̄ = (0, 0, 0, m) for m > 0, reps.
p̄ = (1, 0, 0, 1) for m = 0.
The groups are:
For m > 0 the group SU (2) of unitary 2 × 2 matrices with
determinant 1 (=covering group of SO(3).)
For m = 0 the 2D euclidean group E2 , consisting of matrices of
the form
iθ/2
e
0
1 z
0 1
0
e−iθ/2
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
8 / 73
Induced representations
A representation V of a “little group” induces a representation of the
+ a Lorentz
full group in the following way: Choose for each p ∈ Hm
transformations Λp with
Λp p̄ = p.
A possible choice for the corresponding matrices in SL(2, C) is
1
Ap = √
m
for m > 0 and
Bp =
p0 + p3 p1 − ip2
p1 + ip2 p0 + p3
p
p0 + p3
1
2
p
√ +ip
p0 +p3
0
√
1/2
!
1
p0 +p3
for m = 0. (Λp = Λ(Ap ) is a Lorentz boost, Λp = Λ(Bp ) a combination
of a rotation and a boost.)
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
9 / 73
Then, for every Λ,
−1
−1
−1
Λ−1
p ΛΛΛ−1 p p̄ = Λp ΛΛ p = Λp p = p̄,
i.e., Λ−1
p ΛΛΛ−1 p belongs to the stabilizer group of p̄.
Moreover,
W (p, Λ) := V (Λ−1
p ΛΛΛ−1 p )
satisfies the cocycle relation and
µ
(U (a, Λ)ψ)(p) := eiaµ p W (p, Λ)ψ(Λ−1 p)
↑
is a unitary representation of P+
.
It is irreducible if and only if V is irreducible, and all irreducible
representations are obtained in this way.
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
10 / 73
The irreducible, unitary representations of P+↑
↑
Thus the classification of the irreducible unitary representations of P+
is given in terms of the mass m and the irreducible unitary
representations of SU (2) for m > 0, and of E2 for m = 1. There are
three classes of representations:
[m, s] with m > 0 and the spin s = 0, 21 , 1, . . . , which defines an
irreducible representation of SU (2).
[0, h] with the helicity h = 0, ± 21 , ±1, . . . defining one-dimensional
represesentations of E2 .
[0, Ξ, ±], the mass zero “infinite spin” representations,
corresponding toinfinite
dimensional representations of E2 with
1 z
the translations
nontrivially represented.
0 1
These representations can be obtained as limits of [m, s] with
m → 0, s → ∞ but Ξ2 = m2 s(s + 1) fixed. The helicity spectrum is
h = 0, 1, 2, . . . for [0, Ξ, +] and h = 21 , 23 , 52 , . . . for [0, Ξ, −]
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
11 / 73
Localization I: Problems with position operators
↑
We have seen how the representation theory of P+
naturally leads to
energy and momentum operators as generators of translations in
space-time. Consider the simplest case of a spinless particle, i.e., an
irreducible representation [m, 0]. The Lorentz invariant measure
+ is equivalent to the measure dp/2ω on R3 with
dµm (p) on Hm
p
p
ωp = p2 + m2 .
Thus, the scalar product for the wave functions ψ(p) is
Z
dp
hψ1 , ψ2 i = ψ1 (p)ψ2 (p)
.
2ωp
The momentum operator is
(Pψ)(p) = pψ(p).
What about a position operator?
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
12 / 73
A natural choice is obtained by writing the scalar product as
Z
hψ1 , ψ2 i = ψ̂1 (x)ψ2 (x)dx
which means that
1
ψ̂(x) =
(2π)3/2
Z
ψ(p)
eip·x p
dp.
2ωp
The Newton-Wigner position operator Xnw is now defined by
(Xnw ψ̂)(x) := xψ̂(x).
Its action in momentum space, i.e. on ψ(p), is given by
p
Xnw ψ = i∇p −
ψ.
2(p2 + m2 )
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
13 / 73
The NW operator has the desirable properties of being translationally
covariant and states localized in disjoint sets w.r.t. this operator are
orthogonal. There is, however a problem with causality:
The time evolution of ψ̂ is given by
Z
ψ̂t (x) = Kt (x − x0 )ψ̂0 (x0 )dx0
with
1
Kt (x) =
(2π)3/2
Z
ei(p·x−ωp t) dp.
Relativistic causality would require that the propagator Kt (x) vanishes
for |x| > ct. This, however, is not the case if t 6= 0, because the Fourier
transform of Kt , i.e., e−iωp t , is not an entire analytic function of p.
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
14 / 73
In fact, the conflict between causality and localization via position
operators for particles is a completely general feature of relativistic
quantum physics:
Theorem
Suppose there are is a mapping ∆ 7→ E∆ from subsets of space-like
hyperplanes in Minkowski space into projectors on H such that
(1) U (a)E∆ U (a)−1 = E∆+a
(2) E∆ E∆0 = 0 if ∆, ∆0 space-like separated.
Then E∆ = 0 for all ∆.
Proof. The spectrum condition implies that the function a 7→ U (a, 1)Ψ
has, for every Ψ ∈ H, an analytic continuation into Rd + iV+ ⊂ C4 .
The second condition (2) means that
hE∆ Ψ, U (a)E∆ Ψi = hΨ, E∆ E∆+a U (a)Ψi = 0
on an open set in Minkowski space. But an analytic function that is
continuous on the real boundary of its analyticity domain and vanishing
on an open subset of this boundary vanishes identically. Jakob Yngvason (Uni Vienna)
Local Quantum Physics
15 / 73
Conclusion: Localization in terms of position operators is
incompatible with causality in relativistic quantum physics.
Way out: Shift from localization of ‘wave functions’ to localization of
operators (‘observables’, or ‘operations’). Space-time coordinates
appear as variables of quantum fields. Causality manifests itself in
commutativity (or anticommutativity) of the quantum field operators at
space-like separation.
The dependence of field operators on the coordinates is by necessity
singular and requires smearing with test functions. Localization at a
point is problematic, but localization in an open domain makes sense.
The most flexible general conceptual framework incorporating these
ideas is that of “Local Quantum Physics”(LQP), also called “Algebraic
Quantum Field Theory”(AQFT).
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
16 / 73
Localization II. The framework(s) of Local Quantum
Physics
Ingredients:
Hilbert space of state vector, H
P
Minkowski space M = IR4 , with x · y = x0 y0 − 3j=1 xj yj
↑
Unitary representation U (a, Λ) of the Poincaré group P+
on H
A (unique) invariant state vector Ω ∈ H (vacuum)
A net of algebras F(O) of bounded operators on H, indexed by
regions O ⊂ M with F(O1 ) ⊂ F(O2 ) if O1 ⊂ O2 . The algebras are
assumed to be closed in the weak operator topology.
Requirements:
(Causality) F(O1 ) commutes (or commutes after a twist) with
F(O2 ) if O1 and O2 space-like separated.
(Covariance) U (a, Λ)F(O)U (a, Λ)−1 = F(ΛO + a).
(Spectrum condition) spec U (a, 1) ⊂ V+ .
(Cyclicity of vacuum) ∪O F(O)Ω is dense in H.
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
17 / 73
Remarks:
The operators in F(O) can intuitively be thought of as generating
physical operations (in the sense of K. Kraus) carried out in the
space-time region O.
Usually (but not always!) F(O) is nontrivial for all open regions O.
Associated with the field net {F(O)}O⊂M there is usually another
net of operator algebras, {A(O)}O⊂M , representing local
observables and commuting with the field net and itself at
space-like separations. Usually this is a subnet of the field net,
selected by invariance under some (global) gauge group.
Fundamental insight of Rudolf Haag (1957): Interactions between
particles (emerging asymptotically for larges times), in particular
scattering amplitudes, are already encoded in the field net: It is not
necessary to attach specific interpretations to specific operators in
F(O) besides the localization.
Further fundamental insight (H.J. Borchers, 1965; S. Doplicher, R.
Haag, J. Roberts (1969-90)): The field net {F(O)} and the gauge
group can in principle be derived from the observable net {A(O)}.
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
18 / 73
The Reeh-Schlieder Theorem
Before discussing how nets of local algebras typically arise we
consider a simple, but important general result. For this one additional
assumption is needed, weak additivity: For every fixed open set O0
the union of all translates, F(O0 + x), is dense in the union of all F(O)
in the weak operator topology.
Theorem
F(O)Ω is dense in H for all open sets O, i.e., the vacuum is cyclic
for every single local algebra and not just for their union.
Proof. Write U (a) for U (a, 1). Pick O0 ⊂ O such that O0 + x ⊂ O for all
x with |x| < ε, for some ε > 0. If Ψ ⊥ F(O)Ω, then
hΨ, U (x1 )A1 U (x2 − x1 ) · · · U (xn − xn−1 )An Ωi = 0 for all Ai ∈ O0 and
|xi | < ε. Then use analyticity of U (a) to conclude that this must hold
for all xi . The theorem now follows by appealing to weak additivity. Jakob Yngvason (Uni Vienna)
Local Quantum Physics
19 / 73
Corollary
The vacuum is a separating vector for every O such that its causal
complement O0 contains an open set, i.e., AΩ = 0 for A ∈ F(O)
implies A = 0.
Proof. If O0 ⊂ O0 , then ABΩ = BAΩ = 0 for all B ∈ F(O). But F(O)Ω
is dense, so A = 0. Remark 1. The Reeh-Schlieder Theorem and its Corollary hold, in
fact, for any state vector Ψ0 with bounded energy spectrum and not
only the vacuum, because U (x)Ψ0 is analytic in a whole complex
neighborhood of x for such vectors.
Remark 2. No violation of causality is implied by the Reeh-Schlieder
Theorem. The theorem is ‘just’ a manifestation of unavoidable
correlations in the vacuum state (or any state with bounded energy
spectrum).
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
20 / 73
Quantum fields
The local algebras F(O) and A(O) are in most cases (but not always!)
obtained from relativistic quantum fields. These are functions Φα (x)
on Minkowski space with values in (unbounded) operators on a Hilbert
space and fulfilling some general requirements.
In fact, it turns out that the dependence on x is necessarily singular, so
these objects have to be understood as operator valued
distributions. This means that only averaged operators with some
test functions f ,
Z
Φα (f ) =
Φα (x)f (x)dx
are well defined.
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
21 / 73
Given a quantum field, Φα , the corresponding local algebras F(O) are
usually defined by taking bounded functions of the unbounded field
operators Φα (f )’s with support of f contained in O. This can be
mathematically dicy, but in the simplest situations the Φα (f ) are
(essentially) self-adjoint for real-valued test functions and the F(O)
can be generated by the spectral projectors of the field operators.
More generally, one can consider the polar decomposition of the
operators. A net generated in this way automatically satisfies the
assumption of weak additivity, but one must also prove that it inherits
local (anti)commutativity from the field operators.
A precise definition of the concept “relativistic quantum field”
(encompassing most, but not all, interesting cases) was formulated by
A. Wightman 1956-1964 (partly in collaboration with L. Gårding).
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
22 / 73
The Wightman axioms
The Φα are operator valued distributions and the field operators Φα (f )
can be multiplied freely on a dense domain D in a Hilbert space H.
↑
Moreover, a unitary representation U (a, Λ) of P+
(more precisely, of
ISL(2, C)) is assumed to be given on H, leaving D invariant and with a
unique vacuum state vector Ω ∈ D.
The further assumptions are:
(Causality) [Φα (x), Φβ (y)]∓ = 0 if (x − y) space-like.
(Covariance)
P
U (a, A)Φα (x)U (a, A)−1 = β Φα (Λ(A)x + a)β D(A)βα with a
(finite dimensional, nonunitary) representation D of SL(2, C).
(Spectrum condition) spec U (a, 1) ⊂ V+ .
(Cyclicity of the vacuum) Ω is cyclic for the algebra generated by
the field operators.
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Local Quantum Physics
23 / 73
Denote by P(O) the algebra consisting of all sums of products of field
operators Φα (f ) with supp f ⊂ O. We can write the Wightman axiom
in complete analogy to the previously considered requirements for a
local net of operator algebras F(O):
(Causality) P(O1 ) commutes (or commutes after a twist) with
P(O2 ) if O1 and O2 space-like separated.
(Covariance) U (a, A)P(O)U (a, A)−1 = P(Λ(A)O + a)
(Spectrum condition) spec U (a, 1) ⊂ V+ .
(Cyclicity of vacuum) Ω is cyclic for ∪O P(O).
Differences:
The algebras F(O) consist of bounded operators, P(O) of
unbounded operators (in general).
The algebras F(O) are closed in the weak operator topology.
Further remark: When a local net {F(O)} is generated by a quantum
field, the role of the field is analogous to that of a coordinate system in
differential geometry. In particular, the field is not unique for a given net
{F(O)}. A Borchers class consists of fields generating the same net.
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
24 / 73
The Wightman functions (distributions)
The vacuum expectation values of products of field operators,
Wα1 ,...,αn (x1 , . . . , xn ) = hΩ, Φα1 (x1 ) . . . Φαn (xn )Ωi
are called Wightman functions (more precisely, Wigthman
distributions).
The Wightman functions have an analytic continuation
Wα1 ,...,αn (z1 , . . . , zn ) to a certain domain of (z1 , . . . , zn ) ∈ C4n . In
particular they have a continuation to points with purely imaginary time
components (”euclidean points”). The Wightman functions restricted to
these points are called Schwinger functions.
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
25 / 73
PCT, Spin and Statistics
The analyticity of the Wightman functions is the basis for some famous
general theorems of relativistic quantum field theory:
PCT Theorem:
Every Wightman field has PCT symmetry in the following sense:
There is an anti-unitary operator Θ satisfying
ΘU (a, Λ)Θ−1 = U (−a, Λ)
ΘΦα (x)Θ−1 = cα Φα (−x)∗
Spin-Statistics Theorem:
If Φα transforms with a double valued (spinorial) representation of L↑+
then commutativity for space-like separation is excluded.
Conversely, anticommutativity is excluded if If Φα transforms with a
single valued representation of L↑+ .
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
26 / 73
The Bisognano-Wichmann Theorem
The PCT theorem was used by J. Bisognano and E. Wichmann in
1976 to derive a structural result that is of fundamental importance for
the application of Tomita-Takesaki modular theory in relativistic
quantum field theory.
Let W be a space-like wedge in space-time, i.e., a Poincaré transform
of the standard wedge
W1 = {x ∈ R4 : |x0 | < x1 }.
With W is associated a one-parameter family ΛW (t) of Lorentz boosts
that leave W invariant, and a reflection jW about the edge of the
wedge that maps W into the opposite wedge (causal complement) W 0 .
For the standard wedge W1 this reflection is the product of the
space-time inversion θ and a rotation R(π) by π around the 1-axis. For
a general wedge the corresponding operators are obtained by
combination with the Poincaré transformation that takes W1 to W.
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
27 / 73
The Bisognano Wichmann Theorem relates these geometric
transformations to the closed antilinear operators SW that map the
dense sets P(W)Ω into itself and are defined by
SW AΩ = A∗ Ω
for A ∈ P(W). (If the algebra F(W) of bounded operators is
generated by P (W) the same operator is obtained taking A ∈ F(W).)
It is sufficient to consider W1 and we drop the index W1 for simplicity.
The Operator S ≡ SW1 has a polar decomposition:
S = J∆1/2
with J antiunitary and ∆ = S ∗ S a positive operator.
Bisognano-Wichmann Theorem
J = ΘU (R(π)) and ∆it = U (Λ(2πt)).
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
28 / 73
Free fields
A Wigthman field satisfying the linear Klein Gordon equation
( + m2 )Φα (x) = 0
with = ∂µ ∂ µ is called a free field.
Characteristic for free fields is that the (anti)commutator is a c-number:
[Φα (x), Φβ (y)]∓ = hΩ, [Φα (x), Φβ (y)]∓ Ωi1.
As a consequence all Wightman functions are uniquely determined by
the two point function(s):
Wα1 ,...,α2n (x1 , . . . , x2n ) =
X
Wαi1 ,αj1 (xi1 , xj1 ) · · · Wαin ,αjn (xin , xjn ).
pairings {ik <jk }
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
29 / 73
The two point function, in turn, is essentially uniquely determined by
the covariant transformation property of the field under ISL(2, C).
The structure of the Wightman functions leads to a concrete realization
of the field operators in terms of creation and annihilation operators on
Fock space.
This space is defined as
HFock =
M
⊗
H1 s,a
n
n=0
with H1 the subspace obtained by applying the field operators
(including adjoints) once to the vacuum vector Ω, and ⊗s,a denotes the
symmetric or antisymmetric tensor product.
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
30 / 73
Examples I: The free, neutral scalar field
The base space for the bosonic Fock space (the 1-particle space) is in
this case
H1 = L2 (R4 , dµm (p)) ∼
= L2 (R3 , dp/2ωp ).
↑
It carries the irreducible representation [m, 0] of P+
. For ψ ∈ H1 we
have the creation and annihilation operators a(ψ) and a∗ (ψ)
satisfying the CCR
[a(ψ), a∗ (ϕ)] = hψ, ϕi1.
For a test function f on R4 we define the vector ψf ∈ H1 by restricting
+.
the Fourier transform f˜ to the mass hyperboloid Hm
The field operators are then given by
Φ(f ) = 2−1/2 (a(ψf ) + a∗ (ψf¯)).
The 2-point function is
W2 (x − y) = (2π)
Jakob Yngvason (Uni Vienna)
−3/2
Z
ei(x
0 −y 0 )ω
p
ei(x−y)·p
Local Quantum Physics
dp
=: i∆+
m (x − y).
2ωp
31 / 73
Examples II: Generalized free fields, Wick powers,
in the definition of H1 the measure dµm (p) by
RReplacing
dρ(m2 )dµm (p), where dρ(m2 ) is some positive mass distribution
(“Lehmann weight”), leads to a generalized free field. The structure of
the n-point
functions is the same as for the free fields but now with with
R
W2 = dρ(m2 )i∆+
. The one particle space carries the reducible
R ⊕m
↑
representation
[m, 0]dρ(m2 ) of P+
.
Wick powers, denoted : Φn : (x), of free (or generalized free) fields
can be defined by expanding formally Φ(x)n (which is ill defined) in
creation and annihilation operators and applying the usual Wick
ordering: creators to the left, annihilators to the right. This amounts to
the subtraction of singularities, for instance
: Φ2 : (x) = lim (Φ(x)Φ(y) − hΩ, Φ(x)Φ(y)Ωi) .
y→x
Wick powers of the free field with n odd generate the same local
algebras F(O) as the free field itself.
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
32 / 73
Examples III: The free Dirac field
The base space for the fermionic Fock space is here
H1 = C2 ⊗ L2 (R3 , dp/2ωp ) ⊕ C2 ⊗ L2 (R3 , dp/2ωp) .
It carries the reducible representation [m, 12 ] ⊕ [m, 12 ] of ISL(2, C).
The Dirac field Ψµ is defined in terms of the creation and annihilation
operators for these irreducible subspaces and certain functions that
intertwine between the p-dependent “Wigner rotations” W (p, Λ) and
the representations A 7→ D(1/2, 0) (A) := A and A 7→ D(0, 1/2) (A) := Ā
of SL(2, C), in order to ensure a local transformation law.
The Dirac equation is
(iγ µ ∂µ − m)Ψ = 0
with [γ µ , γ ν ]+ = g µ,ν , and Ψ transforms w.r.t. ISL(2, C) as
X
U (a, A)Ψµ (x)U (a, A)−1 =
Ψν (Λ(A)x + a)D(A)νµ
ν
with D ∼
= D(1/2, 0) ⊕ D(0, 1/2) .
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
33 / 73
The charged sectors of the Dirac field
The Dirac field is an instructive example to illustrate the difference
between a field net {F(O)} and an observable net {A(O)}.
The field operators Ψµ (f ) of the Dirac field are not hermitean and they
do not commute at space-like separation. Hence they do not qualify as
observables, but they are nevertheless needed to generate the whole
Hilbert space.
The Hilbert space is a direct sum of charge sectors
∞
M
H=
H(n) .
n=−∞
Here
H(n)
is an eigenspace of the charge operator
Z
Q = w- lim QR = w- lim
χR (x)j0 (0, x)dx
R→∞
R→∞
with χR the (smoothed) characteristic function of a ball of radius R
and
j0 (x) =: Ψ̄(x)γ0 Ψ(x) :
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Local Quantum Physics
34 / 73
The (global) gauge group U (1) operates on H via eiλQ , and the bilinear
expressions in Ψ,
jµ (x) =: Ψ̄(x)γµ Ψ(x) :
are gauge invariant, hermitian, and commute at space-like separation.
The smeared operators jµ (f ) generate the subnet A(O) ⊂ F(O) of
observables. In contrast to the field net F(O) that is irreducibly
represented on H, the observable net leaves every H(n) invariant and
is thus reducibly represented on H.
Two facts are noteworthy:
The restriction of the operators to each H(n) is a faithful
representation of the observable net.
The restrictions for different n’s define unitarily inequivalent
representations of the net of observables.
The latter follows simply from the fact that QR is an observable for all
finite R that converges weakly to n1 on H(n) when R → ∞. This
0
cannot hold if QR | H(n) = V −1 QR V with V : H(n) → H(n ) , n 6= n0 .
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
35 / 73
Interacting fields
The examples discussed so far do not describe interactions (the
scattering matrix, when defined, is the identity). They are, however, still
of fundamental importance for (at least) the following reasons:
Free fields describe the states of free particles that emerge
asymptotically for t → ±∞ in theories with interaction.
Free fields are often the starting point for the introduction of
interactions.
Generalized free fields and Wick powers can be used to illustrate
various general properties and test conjectures about relativistic
quantum fields.
But an important question remains: How can fields with interactions
be constructed?
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
36 / 73
Construction Methods
Lagrangian field theory plus canonical quantization plus
perturbation theory plus renormalization. Leads rigorously to
theories with interaction defined in terms of formal power series in
a coupling constant in all space-time dimensions.
Constructive QFT (J. Glimm, A. Jaffe and followers) has produced
interacting field theories in space-time dimensions 1+1 and 1+2,
but not yet in 1+3.
Conformal QFT in 1+1 space-time dimensions based on Virasoro
algebras etc.
Big challenge in QFT: Find new methods of construction and
classification! Recent progress through deformations of known models
and use of modular theory.
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
37 / 73
Modular Localization
In 2002 R. Brunetti, D. Guido and R. Longo introduced the concept of
modular localization that is based on a certain converse of the
Bisognano-Wichmann Theorem (1976).
Recall first the BW result: Let W be a space-like wedge in space-time,
i.e., a Poincaré transform of the standard wedge
W1 = {x ∈ R4 : |x0 | < x1 }.
With W is associated a one-parameter family ΛW (t) of Lorentz boosts
that leave W invariant and a reflection jW about the edge of the wedge
that maps W into the opposite wedge (causal complement) W 0 . For
the standard wedge W1 this is the product of the space-time inversion
θ and a rotation R(π) by π around the 1-axis. For a general wedge the
corresponding operators are obtained by combination with the
Poincaré transformation that takes W1 to W.
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Local Quantum Physics
38 / 73
The Bisognano Wichmann theorem relates these geometric
transformations to the closed antilinear Operators SW that map the
dense sets P(W)Ω into itself and are defined by
SW AΩ = A∗ Ω
for A ∈ P(W). (If the algebra F(W) of bounded operators is generated
by P (W) the same operator is obtained taking A ∈ F(W).
It is sufficient to consider W1 and we drop the index W1 for simplicity.
The Operator S ≡ SW1 has a polar decomposition:
S = J∆1/2
with J antiunitary and ∆ = S ∗ S a positive operator.
Bisognano-Wichmann Theorem
J = ΘU (R(π)) and
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∆it = U (Λ(2πt))
Local Quantum Physics
39 / 73
We now want to invert this procedure to construct a local net, starting
↑
from a representation of P+ , which is P+
augmented by space-time
reflection.
Let U1 be an (anti)unitary representation of P+ satisfying the spectrum
condition on a Hilbert space H1 .
For a given wedge W, let ∆W be the unique positive operator
satisfying
∆itW = U1 (ΛW (t)) , t ∈ R ,
and let JW to be the anti–unitary involution representing jW .
Define
1/2
SW := JW ∆W .
This operator is densely defined on H(1) , closed, antilinear and
involutive.
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Local Quantum Physics
40 / 73
Define
K(W) := {φ ∈ domain SW : SW φ = φ} ⊂ H1 .
This space satisfies:
K(W) is a closed real subspace of H1 in the real scalar product
Re h·, ·i.
K(W) ∩ iK(W) = {0}.
K(W) + iK(W) is dense in H1 .
Such real subspaces of a complex Hilbert space are called standard in
Tomita-Takesaki theory.
Moreover
K(W)⊥ := {ψ : Im hψ, φi = 0 for all φ ∈ K(W)} = K(W 0 )
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Local Quantum Physics
41 / 73
The functorial procedure of second quantization leads for any ψ ∈ H1
to an (unbounded) field operator Φ(ψ) on the Fock space
M∞
⊗
H=
H1 symm
n=0
such that
[Φ(ψ), Φ(φ)] = i Im hψ, φi1.
Hence
[Φ(ψ1 ), Φ(ψ2 )] = 0
if ψ1 ∈ K(W), ψ2 ∈ K(W 0 ).
Finally, a causal net of algebras F(O) is defined by
F(O) := {exp(iΦ(ψ)) : ψ ∈ ∩W⊃O K(W)}00 .
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
42 / 73
Although this construction produces only interaction free fields it is
remarkable for at least two reasons:
It uses as sole input a representation of the Poincaré group, i.e., it
is intrinsically quantum mechanical and not based on any
“quantization” of a classical theory.
It works for the irreducible representation of the Poincaré group of
any mass and spin or helicity, including the zero mass, infinite spin
representations, that can not be generated by point localized
fields, i.e., operator valued distributions à la Wightman (JY,1970).
Alternatively, the algebras can be described in terms of string
localized fields Φ(x, e) (J. Mund, B. Schroer, JY, 2004-2006) with e
a space like vector of length 1, and [Φ(x, e), Φ(x, e)] = 0 if the
“strings”(rays) x + λe and x0 + λ0 e0 with λ, λ0 > 0 are space-like
separated.
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
43 / 73
Remarks:
Free string localized fields can be constructed for all irreducible
represenations of the Poincaré group and their most general form
is understood. They show a better UV behavior than Wightman
fields. One can also a construct string localized vector potential for
the EM field, and there are generalizations to massless fields of
arbitrary helicities.
Localization in cones rather than bounded regions occurs also in other
contexts:
Fields generating massive particle states can always be localized
in space-like cones (D. Buchholz, K. Fredenhagen 1982) and in
massive gauge theories no better localization is expected.
In QED localization in light-like cones may be expected (D.
Buchholz, J. Roberts 2010-2013).
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
44 / 73
The Structure of Local Algebras
Some terminology:
B(H) the algebra of all bounded, linear operators on a Hilbert space H.
A ⊂ B(H) subalgebra.
Commutant:
A0 = {B ∈ B(H) : [A, B] = 0 for all A ∈ A}.
von Neumann algebra:
A = A00 .
States: A (normal) state on a von Neumann algebra A is a positive
linear functional of the form ω(A) = trace (ρA), ρ ≥ 0, trace ρ = 1.
Pure state: ω = 12 ω1 + 21 ω2 implies ω1 = ω2 = ω. Beware: If A =
6 B(H),
ρ is not unique and pure state is not the same as “vector state”!
A vector ψ ∈ H is cyclic for A if Aψ is dense in H and separating if it
is cyclic for A0 . (Equivalent: Aψ = 0 for A ∈ A implies A = 0.)
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Local Quantum Physics
45 / 73
A factor is a v.N. algebra A such that
A ∨ A0 ≡ A ∪ A0
00
= B(H)
which is equivalent to
A ∩ A0 = C1.
Motivation from QM: Division of a system into two subsystems.
Simplest case (familiar from QIT):
H = H1 ⊗ H 2
A = B(H1 ) ⊗ 1 ,
A0 = 1 ⊗ B(H2 )
This is the Type I case. It is characterized by the existence of minimal
projectors in A: If ψ ∈ H1 , Eψ = |ψihψ|, then
E = Eψ ⊗ 1 ∈ A
is a minimal projector, i.e., it has no proper subprojectors in A.
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
46 / 73
The other extreme is the Type III case:
For every projector E ∈ A there exists an isometry W ∈ A with
W ∗W = 1 ,
WW∗ = E
For a type III factor, A ∨ A0 is not a tensor product factorization.
Need we bother about other cases than type I in quantum
physics?
Fact: In LQP the local algebras of observables A(O) are in all known
cases of type III.
More precisely, under some reasonable assumptions, they are
isomorphic to the unique, hyperfinite type III1 factor in a finer
classification due to A. Connes. This classification is based on
Tomita-Takesaki modular theory.
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Local Quantum Physics
47 / 73
Some consequences of the type III property
1. Absence of pure states:
A type III factor A has no pure (normal) states, i.e., for every ω there
are ω1 and ω2 , different from ω, such that
ω(A) = 12 ω1 (A) + 12 ω2 (A)
for all A ∈ A. (Consequence for interpretation of mixtures!)
On the other hand every state on A is a vector state, i.e., for every ω
there is a (non-unique!) ψω ∈ H such that
ω(A) = hψω , Aψω i
for all A ∈ A.
(Remark. There is a distinguished choice of ψω provided by modular
theory: ψω is unique if chosen from the “positive cone” of A.)
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
48 / 73
A type III factor shares these features with the commutative von
Neumann algebra L∞ (R, dx) of bounded, measurable functions on R,
considered as multiplication operators on H = L2 (R, dx). Its projectors
correspond to sets of positive measure and there are no minimal
projectors. On the other hand, every normal state is given a positive
function in L1 (R, dx), i.e., the square of a function in L2 (R, dx).
Concrete examples of type III factors can be defined in terms of infinite
tensor products of matrix algebras (“spin chains”) and, besides in
quantum field theory, they arise naturally in the thermodynamic limit of
non relativistic systems at nonzero temperature.
In LQP the occurrence of type III algebras is a short distance (high
energy) effect. Its general derivation relies on:
The Bisognano Wichmann Theorem for the wedge algebras
A(W ), that identifies the modular group w.r.t. the vacuum with a
geometric transformation (Lorentz-boosts) whose spectrum is the
whole of R.
An assumption about scaling limits that allows to carry the
argument for wedge algebras over to double cone algebras.
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
49 / 73
A concrete type IIIλ factor with 0 < λ < 1 is generated by the infinite
power of the algebra M2 (C) of complex 2 × 2 matrices in the GNS
representation defined by the state
Y
ω(A1 ⊗ A2 ⊗ · · · ⊗ AN ⊗ 1 · · · ) =
tr(An ρλ )
n
with An ∈ M2 (C), 0 < λ < 1 and
1
ρλ =
1+λ
1 0
.
0 λ
A type III1 factor is obtained from an analogous formula for the infinite
product of complex 3 × 3 matrices in the representation defined by
tracing with the matrix


1 0 0
1
0 λ 0 
ρλ,µ =
1+λ+µ
0 0 µ
where λ, µ > 0 are such that
Jakob Yngvason (Uni Vienna)
log λ
log µ
∈
/ Q.
Local Quantum Physics
50 / 73
2. Local preparability of states:
For every projector E ∈ A(O) there is an isometry W ∈ A(O) such that
E = W W ∗ but W ∗ W = 1. This implies that for any state ω we have
ωW (E) = 1
but
ωW (B) = ω(B) for B ∈ A(O0 )
where ωW ( · ) := ω(W ∗ · W ).
In words: Every state can be changed into an eigenstate of any given
local projector, by a local operation that is independent of the state and
does not affect the state in the causal complement.
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
51 / 73
3. Local comparison of states cannot be achieved by means of
positive operators:
For O ⊂ M and two states ϕ and ω define their local difference by
DO (ϕ, ω) ≡ sup{|ϕ(A) − ω(A)| : A ∈ A(O), kAk ≤ 1}
In a type I algebra local differences can, for a dense set of states, be
tested in the following sense by means of positive operators:
For a dense set of states ϕ there is a positive operator Pϕ,O such that
DO (ϕ, ω) = 0 if and only if ω(Pϕ,O ) = 0.
For a type III algebra, on the other hand, such operators do not exist
for any state.
Failure to recognize this has in the past led to spurious ‘causality
problems’, inferred from the fact that for a positive operator P an
expectation value ω(eiHt P e−iHt ) with H ≥ 0 cannot vanish in an
interval of t’s without vanishing identically. (Diskussions about “Fermi’s
two atom system”.)
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
52 / 73
The use of approximate theories
The last point leads to the following general remarks:
Constructions of fully relativistic models for various phenomena where
interactions play a decisive role are usually very hard (and even
impossible) to carry out in practice.
Hence one one must as a rule be content with some approximations
(divergent perturbation series without estimates of error terms), or
semi-relativistic models with various cut-offs (usually at high energies).
Such models usually violate one or more of the general assumptions
of LQP.
Computations based on such models may well lead to results that are
in conflict with basic principles of relativistic quantum physics such as
an upper limit for the propagation speed of causal influence, but this is
quite understandable and should not be the the cause of worry (or of
unfounded claims) once the reason is understood.
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
53 / 73
Entanglement in LQP
If A1 and A2 commute, a state ω on A1 ∨ A2 is by definition
entangled, if it can not be approximated by convex combinations of
product states on A1 ∨ A2 .
General result: If A1 and A2
commute
are non-abelian
possess each a cyclic vector
A1 ∨ A2 has a a separating vector
then the entangled states form a dense, open subset of the set of all
states.
This applies directly to the local algebras of LQP because of the
Reeh-Schlieder Theorem that says that every analytic vector for the
energy (in particular the vacuum) is cyclic and separating for the local
algebras.
Thus the entangled states on A(O1 ) ∨ A(O2 ) are generic for space-like
separated, bounded open sets O1 and O2 .
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
54 / 73
The type III property implies even stronger entanglement:
If A is a type III factor, then A ∨ A0 does not have any product states,
i.e., all states are entangled for the pair A, A0 .
Haag duality means by definition that A(O)0 = A(O0 ). Thus, if Haag
duality holds, a quantum field in a bounded space-time region can
never be disentangled from the field in the causal complement.
By allowing a small distance between the regions, however,
disentanglement is possible, provided the theory has the split
property, that mitigates to some extent the ‘rigidity’ implied by the type
III character of the local algebras.
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
55 / 73
Causal Independence and Split Property
A pair of commuting von Neumann algebras, A1 and A2 , in a common
B(H) is causally (statistically) independent if for every pair of states,
ω1 on A1 and ω2 on A2 , there is a state ω on A1 ∨ A2 such that
ω(AB) = ω1 (A)ω2 (B)
for A ∈ A1 , B ∈ A2 .
In other words: States can be independently prescribed on A1 and A2
and extended to a common, uncorrelated state on the joint algebra.
This is really the von Neumann concept of independent systems.
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
56 / 73
The Split property for commuting algebras A1 , A2 means, by
definition, the following: There is a type I factor N such that
A1 ⊂ N ⊂ A02
which again means: There is a tensor product decomposition
H = H1 ⊗ H2 such that
A1 ⊂ B(H1 ) ⊗ 1 ,
A2 ⊂ 1 ⊗ B(H2 ).
In the field theoretic context causal independence and split
property are equivalent.
The split property for local algebras separated by a finite distance can
be derived from a condition (“nuclearity”) that expresses the idea that
the local energy level density (measured in a suitable sense) does
not increase too fast with the energy (Buchholz and Wichmann,
1986).
Nuclearity is not fulfilled in all models (some generalized free fields
provide counterexamples), but it is a reasonable requirement
nevertheless.
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
57 / 73
The split property together with the type III property of the strictly local
algebras leads to a strong version of the local preparability of
states:
Theorem (Strong local preparability)
For every state ϕ (‘target state’) and every bounded O there is an
isometry W ∈ A(Oε ) (with Oε slightly larger than O) such that for an
arbitrary state ω (‘input state’)
ωW (AB) = ϕ(A)ω(B)
for A ∈ A(O), B ∈ A(Oε )0 , where ωW ( · ) = ω(W ∗ · W ).
In particular, ωW is uncorrelated and its restriction to A(O) is the target
state ϕ, while in the causal complement of Oε the preparation has no
effect on the input state ω. Moreower, W depends only on the target
state and not on the input state.
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
58 / 73
Proof: The split property implies that we can write A(O) ⊂ B(H1 ) ⊗ 1,
A(Oε )0 ⊂ 1 ⊗ B(H2 ).
By the type III property of A(O) we have ϕ(A) = hξ, Aξi for A ∈ A(O),
with ξ = ξ1 ⊗ ξ2 . (The latter because every state is a vector state, and
we can regard A(O) as a subalgebra of B(H1 ). ) Then
E := Eξ1 ⊗ 1 ∈ A(Oε )00 = A(Oε ).
By the type III property of A(Oε ) there is a W ∈ A(Oε ) with W W ∗ = E,
W ∗ W = 1. The second equality implies ω(W ∗ BW ) = ω(B) for
B ∈ A(Oε )0 .
On the other hand, EAE = ϕ(A)E for A ∈ A(O) and multiplying this
equation from left with W ∗ and right with W , one obtains
W ∗ AW = ϕ(A)1
by employing E = W W ∗ and W ∗ W = 1. Hence
ω(W ∗ ABW ) = ω(W ∗ AW B) = ϕ(A)ω(B).
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
59 / 73
The Theorem implies also that any state on A(O) ∨ A(Oε )0 can be
disentangled by a local operation in A(Oε ):
Given a state ω on A(O) ∨ A(Oε )0 there is, by the preceding Theorem,
an isometry W ∈ A(O ) such that
ωW (AB) = ω(A)ω(B) .
for all A ∈ A(O), B ∈ A(Oε0 ).
In particular: Leaving a ‘security margin’ between a bounded domain
and its causal complement, the global vacuum state ω(·) = hΩ, · Ωi can
be disentangled by a local operation producing an uncorrelated state
on A(O) ∨ A(Oε )0 that is identical to the vacuum state on each of the
factors.
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
60 / 73
Further issues in relativistic entanglement not discussed here
Entanglement measures for local algebras (in particular Bell-type
correlations, entanglement entropy,. . . )
‘Area laws’ for entanglement entropy implying that entanglement
in states of finite energy is essentially a boundary effect
Mixing of of ‘internal’ and ‘translational’ degrees of freedom that
become entangled under Lorentz boosts (because ‘Wigner
rotations’ depend on the momentum).
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
61 / 73
Some conclusions of the discussion so far
The framework of LQP leads to a special sight on the concepts
‘system’, ‘subsystem’ and ‘particle’: The system is composed of
quantum fields in space-time, represented by a net of local
algebras. A subsystem is represented by one of the local
algebras, i.e., the fields in a specified part of space-time. ‘Particle’
is a derived concept that (for theories with interaction) emerges
asymptotically at large times but is usually not strictly defined at
finite times.
The fact that local algebras have no pure states is relevant for
interpretations of the state concept (attribute of an individual
system or of an ensemble?)
The type III property is relevant for causality issues and local
preparability of states, and responsible for ‘deeply entrenched’
entanglement, that is, however, mitigated by the split property.
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
62 / 73
On the other hand, the framework of LQP does not per se resolve the
riddles of QM. EPR is not explained away by type III factors!
Moreover, the terminology has still an anthropocentric ring
(‘observables’, ‘operations’) as usual in QM.
This is disturbing since physics is concerned with more than controlled
experimenting. We use quantum (field) theories to understand
processes in the interiors of stars in remote galaxies billions of years
ago, or even the ‘quantum fluctuations’ that are allegedly responsible
for fine irregularities in the 3K background radiation. In none of these
cases ‘observers’ were around to ‘prepare states’ or ‘reduce wave
packets’ !
Need a fuller understanding of the emergence of macroscopic ‘effects’.
‘Consistent histories’ and ‘decoherence’ point in the right direction, but
are probably not the last word.
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
63 / 73
Scattering theory
Local Quantum Physics has its roots in the pioneering ideas of Rudolf
Haag from the 1950’s about scattering theory. Hence it is appropriate
to end this survey of LQP by giving a brief account of this theory in a
modern garment, incorporating some ideas of Klaus Hepp (1965).
Consider first non-relativistic scattering theory of n identical bosons
that interact with a two-body potential v. The free Hamiltonian is
H0 =
n
X
(−∇i )2
i=1
while the full Hamiltonian, including the interaction, is
X
H = H0 +
v(xi − xj ).
i<j
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
64 / 73
We assume for simplicity that there are no bound states. Then, if the
potential v decreases sufficiently fast at infinity, it is reasonable to
expect that for any given 1-particle states ψ1 , . . . , ψn there are state
vectors, denoted
in
in
ψ1 × · · · × ψn
out
out
and ψ1 × · · · × ψn
that for t → ∞ and t → +∞ respectively look more and more like the
free state vectors
e−itH0 (ψ1 ⊗s · · · ⊗s ψn ).
More precisely, we expect that
ex
ex
lim ke−itH (ψ1 × · · · × ψn ) − e−itH0 (ψ1 ⊗s · · · ⊗s ψn )k = 0
t→±∞
ex
in
out
where we have written × collectively for × or × .
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
65 / 73
Equivalently, the scattering states, i.e., the states that behave
asymptotically like a system of freely moving particles, are given by
ex
ex
ψ1 × · · · × ψn = s- lim eitH e−itH0 (ψ1 ⊗s · · · ⊗s ψn ).
t→±∞
The existence of the Møller operators (“wave operators”)
W± := s- lim eitH e−itH0
t→±∞
is guaranteed, for instance, if v is square integrable (Cook’s Theorem).
When trying to extend these ideas to relativistic quantum field theory
one encounters the problem that there is in general no splitting of
the Hamiltonian H into a “free” part H0 and an interacting part.
Moreover, the vector ψ1 ⊗s · · · ⊗s ψn need not be defined in the
Hilbert space (except for free fields).
The task is to define a substitute for e−itH0 (ψ1 ⊗s · · · ⊗s ψn ) and prove
that eitH times this substitute converges for |t| → ∞ to a state vector
that can be interpreted in terms of freely moving particles with
wave function ψ1 , . . . , ψn .
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
66 / 73
For simplicity we assume that the energy momentum spectrum has an
+ with m > 0 and that the representation
isolated mass hyperboloid Hm
↑
of P+ on the corresponding Hilbert space H1 is irreducible with spin 0.
Since the vacuum is cyclic for {F(O)} every vector in H1 can be
approximated arbitrarily well by applying local operators to Ω. One can
even define almost local particle creators that generate a dense set
in H1 without any contribution in H1⊥ :
Let h be a test function in the Schwartz space S(R4 ) such that the
support of its Fourier transform does not intersect the complement of
+ . If A ∈ F(O) for some bounded O and
Hm
Z
B := U (x)AU (−x)h(x)dx,
then
BΩ = h̃(P )AΩ ∈ H1 ,
B ∗ Ω = 0.
Moreover, B is almost local in the sense that for every λ > 0 there is a
Bλ ∈ F(λO) such that for λ → ∞
kB − Bλ k → 0
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Local Quantum Physics
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Let f ∈ S(R3 ) be a test function of the space variable x such that its
Fourier transform f˜ has compact support in the momentum variable p.
With B an almost local 1-particle generator we define
Z
U (0, x)BU (0, x)−1 f (x) dx
B(f ) =
R3
Then B(f )Ω = f˜(P)BΩ is a 1-particle state with momentum support
contained in in supp f˜.
The next point to notice its that the function
Z
dp
ft (x) = ei(p·x−ωp t) f˜(p)
2ωp
is a smooth solution of the Klein-Gordon equation
2
∂
2
−
∇
ft (x) = 0.
∂t2
It can be shown to vanish rapidly outside the velocity cone
Cf = {(t, x) : x = tp/ωp , p ∈ supp f + Nε }.
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
68 / 73
With B1 , . . . Bn almost local 1-particle generators and f1,t , . . . fn,t
smooth solutions of the Klein-Gordon equation the state vector
B1 (f1,t ) · · · Bn (fn,t )Ω
is our substitute for e−itH0 (ψ1 ⊗s · · · ⊗s ψn ).
Moreover,
B1 (f1 , t) · · · Bn (fn , t)Ω
with
B(f, t) := eitH B(ft )e−itH
corresponds to eitH e−itH0 (ψ1 ⊗s · · · ⊗s ψn ).
Note that ψ = B(f, t)Ω ∈ H is independent of t, i.e.,
d
B(f, t)Ω = 0.
dt
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
69 / 73
It is now convenient to consider first the case that the supports of the
Fourier transforms f˜i are non-overlapping. In this case the
intersections of the velocity cones Cfi with the hyperplane x0 = t
become nicely separated for sufficiently large |t| and the following
theorem has a fairly simple proof:
Theorem (Scattering states for non overlapping velocities)
If the supports of the Fourier transforms f˜i are non-overlapping, then
ex
ex
ψ1 × · · · × ψn := s- lim B1 (f1 , t) · · · Bn (fn , t)Ω
t→±∞
exists and depends only on the ψi = B(fi )Ω.
For the proof one considers the time derivative
X
d
d
B1 (f1 , t) · · · Bn (fn , t)Ω =
B1 (f1 , t) · · · Bi (fi , t) · · · Bn (fn , t)Ω
dt
dt
i
and shows that it tends rapidly to zero if |t| → ∞.
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
70 / 73
Indeed, the sum can be written as
XX
i
because
B1 (f1 , t) · · ·
j≥i
d
dt B(fi , t)Ω
d
Bj (fj , t), Bj+1 (fj+1 , t) · · · Bn (fn , t)Ω
dt
= 0 for all i.
Due to the non-overlapping velocity supports the commutators
decrease rapidly in norm as |t| → ∞. The convergence of the vector is
then ensured by Cauchy’s criterion. The independence of the choice of
the Bi (fi ) to generate the 1-particle stets ψi is also proved from the
rapid decay of commutators. Jakob Yngvason (Uni Vienna)
Local Quantum Physics
71 / 73
In order to show that the scattering states
ex
ex
ψ1 × · · · × ψn
can be interpreted as expected, i.e, that they describe asymptotically
for large |t| a free collection of free particles, one has to prove:
The states have the right Fock space structure, i.e.,
X
in
in
in
in
hψ1 × · · · × ψn , ϕ1 × · · · × ϕn i =
hψ1 , ϕπ1 i · · · hψn , ϕπn i
π
and likewise for the outgoing states.
For late |t| observations in the hyperplane x0 = t essentially only
detect the states ψi,t in the corresponding velocity cones and the
vacuum outside these cones.
Both points are established with the help of the cluster factorization
property that is a consequence of the uniqueness of the vacuum and
says that
lim hΨ, U (λa)Φi = hΨ, ΩihΩ, Φi
λ→∞
for all space like a (exponentially fast due to the mass gap).
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
72 / 73
The multi-particle states generated by 1-particle states with
non-overlapping velocity distributions span a dense state in Fock
space. Hence, scattering states for overlapping velocities can simply
be obtained by continuity.
in
out
For theories with interactions, × =
6 × so we have two subspaces
Hin ⊂ H and Hout ⊂ H, each with its own Fock space structure, and
two Møller operators W±∗ : Hex → H. Asymptotic completeness
means that
Hin = Hout = H
but note that the Fock space structures are in general different even
when Hin and Hout coincide as sets.
The S-operator (“S-matrix”) is defined through
out
out
in
in
S(ψ1 × · · · × ψn ) = ψ1 × · · · × ψn
and asymptotic completeness is equivalent to unitarity of S.
Jakob Yngvason (Uni Vienna)
Local Quantum Physics
73 / 73