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Lorentz invariance
When we say theory is Lorentz invariant we mean it is invariant
under proper orthochronous subgroup only (those that can be
obtained by compounding ILTs)
based on S-2
Lorentz transformation (linear, homogeneous change of coordinates):
that preserves the interval
Transformations that take us out of proper orthochronous
subgroup are parity and time reversal:
:
orthochronous but improper
All Lorentz transformations form a group:
product of 2 LT is another LT
nonorthochronous and improper
identity transformation:
inverse:
A quantum field theory doesn’t have to be invariant under P or T.
for inverse can be used to prove:
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Infinitesimal Lorentz transformation:
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How do operators and quantum fields transform?
Lorentz transformation (proper, orthochronous) is represented
by a unitary operator
that must obey the composition rule:
thus there are 6 independent ILTs: 3 rotations and 3 boosts
not all LT can be obtained by compounding ILTs!
infinitesimal transformation can be written as:
+1 proper
are hermitian operators = generators of the Lorentz group
-1 improper
from
proper LTs form a subgroup of Lorentz group; ILTs are proper!
using
and expanding both sides, keeping only linear terms in
Another subgroup - orthochronous LTs,
since
we get:
are arbitrary
ILTs are orthochronous!
general rule: each vector index undergoes its own Lorentz transformation!
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Finally, let’s look at transformation of a quantum scalar field:
using
and expanding to linear order in
Recall time evolution in Heisenberg picture:
we get:
this is generalized to:
These comm. relations specify the Lie algebra of the Lorentz group.
P x = P µ xµ = P · x − Ht
We can identify components of the angular momentum and boost
operators:
x is just a label
we can write the same formula for x-a:
e+iP a/! e−iP x/! φ(0)e+iP x/! e−iP a/! = φ(x − a)
we define space-time translation operator:
and find:
and obtain:
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in a similar way for the energy-momentum four vector
P µ = (H/c, P i ) we find:
using
and expanding to linear order in
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Similarly:
we get:
Derivatives carry vector indices:
or in components:
in addition:
is Lorentz invariant
Comm. relations for J, K, P, H form the Lie algebra of the Poincare group.
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Canonical quantization of scalar fields
Relativistic generalization
based on S-3
Hamiltonian for free relativistic particles:
Hamiltonian for free nonrelativistic particles:
spin zero, but can be either bosons or fermions
Furier transform:
a(x) =
!
d3 p
eip·x ã(p) !
3/2
(2π)
d3 x
(2π)3
we get:
Is this theory Lorentz invariant?
eip·x = δ 3 (p)
Let’s prove it from a different direction, direction that we will use for
any quantum field theory from now:
start from a Lorentz invariant lagrangian or action
derive equation of motion (for scalar fields it is K.-G. equation)
can go back to x using:
!
find solutions of equation of motion
3
d p ip·x
e
= δ 3 (x)
(2π)3
show the Hamiltonian is the same as the one above
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Canonical quantization of scalar fields
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A theory is described by an action:
where
(Anti)commutation relations:
is the lagrangian.
Equations of motion should be local, and so
where
is the lagrangian density.
Thus:
[A, B]∓ = AB ∓ BA
Vacuum is annihilated by
is Lorentz invariant:
:
is a state of momentum
is eigenstate of
, eigenstate of
For the action to be invariant we need:
with
with energy eigenvalue:
the lagrangian density must be a Lorentz scalar!
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Any polynomial of a scalar field is a Lorentz scalar and so are products of
derivatives with all indices contracted.
arbitrary constant
Let’s consider:
real solutions:
! = 1, c = 1
and let’s find the equation of motion, Euler-Lagrange equation:
k −→ −k
(we find eq. of motion from variation of an action: making an infinitesimal
variation
in
and requiring the variation of the action to vanish)
thus we get:
integration by parts,
and δφ(x) = 0 at
infinity in any direction
(including time)
is arbitrary function of x and so the equation of motion is
Klein-Gordon equation
(such a
k µ is said to be on the mass shell)
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Solutions of the Klein-Gordon equation:
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Finally let’s choose
one classical solution is a plane wave:
so that
is Lorentz invariant:
manifestly invariant under orthochronous Lorentz transformations
on the other hand
is arbitrary real wave vector and
The general classical solution of K-G equation:
sum over zeros of g, in our case the only zero is k 0 = ω
where
and
are arbitrary functions of , and
is a function of |k| (introduced for later convenience)
for any
the differential
it is convenient to take
the Lorentz invariant differential is:
if we tried to interpret
as a quantum wave function, the second term
would represent contributions with negative energy to the wave function!
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is Lorentz invariant
for which
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In our case:
Finally we have a real classical solution of the K.-G. equation:
,
where again:
For later use we can express
,
in terms of
:
Inserting
where
and we will call
Note,
we get:
.
is time independent.
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Constructing the hamiltonian:
Recall, in classical mechanics, starting with lagrangian
as a function
of coordinates
and their time derivatives
we define conjugate
momenta
and the hamiltonian is then given as:
!
In field theory:
d3 x ip·x
e
= δ 3 (p)
(2π)3
hamiltonian density
and the hamiltonian is given as:
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Summary:
From classical to quantum (canonical quantization):
“coordinates” and “momenta” are promoted to operators satisfying canonical
commutation relations:
is equivalent to:
operators are taken at equal times in the Heisenberg picture
for:
We have rederived the hamiltonian of free relativistic bosons by
quantization of a scalar field whose equation of motion is the KleinGordon equation (starting with manifestly Lorentz invariant lagrangian).
does not work for fermions, anticommutators lead to trivial hamiltonian!
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We have derived the classical hamiltonian:
We kept ordering of a’s unchanged, so that we can easily generalize it to quantum
theory where classical functions will become operators that may not commute.
The hamiltonian of the quantum theory:
(2π)3 δ 3 (0) = V
see the formula for delta function
is the total zero point energy per unit volume
we are free to choose:
the ground state has zero energy eigenvalue.
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