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3-4-3 Time Reversal
Classically the meaning of time-reversal in variance is clear. By reversing the velocities in what used to be the final configuration, a system retraces its way back to some original configuration, if the fundamental dynamics enjoys this invariance. Hence initial and final states are interchanged with identical positions and opposite velocities. This interchange has the consequence that in quantum mechanics the
QUANTIZATION-FREE FIELDS
corresponding operator :T is antiunitary, that is,
The requirement of invariance under :T means that
:TH:Tt
if H is the hamiltonian. We assume time-translation in variance throughout. Consequently, The transition amplitude from the state IqJi) at time ti to the state IqJ f) at time t f is equal to the corresponding amplitude from statel:T qJf) at time ti to I:T qJ;) at time tf. Indeed,
<qJfl e-iH(tf-t,) IqJi) = <:T qJf (:T e-iH(tf-t,) :Tt) I:T qJi)*
= <:TqJfl e-iH(t,-tf) l:TqJi)* = <:TqJ;j e-iH(tf-t,) l:TqJf)
(3-191)
Since momenta, but not positions, are reversed, orbital angular momenta change sign under :T, and so must spins. In particular, helicities are unchanged. For a scalar relativistic quantum field qJ satisfying the Klein-Gordon equation, :T qJ(t, x):Tt equals qJ( - t, x), possibly up to a sign (since qJ is hermitian). This leads to
:T a(k):T t
= I'/a(k)
1'/ =
:T qJ(t, x):Tt = I'/qJ( - t, x)
(3-192)
For the electromagnetic potential the corresponding transformation is (3-193) Consider finally a spinor field, and let us choose the helicity basis. The requirement is :Tbt(k, e):Tt = I'/}bt(k, e) ei8b(k,e) (3-194)
:Tdt(k,e):Tt = I'/Tdt(k,e)e-i8ik,e)
with I'/T a fixed phase. We want to choose eb(k, e) and ed(k, e) in such a way that :Tl/J(x):T t satisfies the time-reversed Dirac equation. From the antilinear character of:T we obtain .
:Tl/J(x):T t = I'/T
fdk~
[b(k,e)u*(k,e)e- i8b (k,e)e ik-x
+ dt(k,e)v*(k,e)e -i8ik,e)e-ik'x]
k in the integral.
(3-195)
The spinors u and v are defined in (3-184). We change k into
If there exists a fixed matrix A such that
Au(k, e) = e- i8b (k,e) u*(k, e) Av(k, e) =
e- i8.(k,e)v*(k, e)
QUANTUM FIELD THEORY
then we shall have
YIjJ(t, x)yt = rfTAIjJ( -t, x)
(3-196)
From the fact that yO is hermitian and l' antihermitian, it follows that
(~T _ m)u*(k, B) = 0
+ m)v*(k, B) =
Multiplication by ySC yields
(~ - m)ysCu*(k, B)
+ m)ySCv*(k, B) =
Consequently, we expect that A -1 is equal to ySC up to a phase provided that the two component spinors CP (k) and CP ( -k) are correctly related. Indeed, in the standard representation of y matrices ySCU*(k,B)=YsC J2m(kO
~*+m (cp:(-k)) + m) 0
+ m)
(- iaz 0 0 ) (cp:( - k)) - iaz 0
~ +m
- J2m(kO
Therefore all that is required is that - iazcp:( - k) be equal to CPe(k) up to a phase. This is the case since - iaz is unitary and (a'k)[ -iazcp:(-k)] = -iaz[ -(a'k)*cp:(-k)] = B[ -iazcp:(-k)] Hence - iazcp:( - k) and yS CU*(k, B)
eiO.(k,e) cp.(k) eill.(k,e) u(k, B)
Now (ySC)-l = -ySC; thus, up to a factor which can be absorbed in rfT we have indeed found the matrix A required in Eq. (3-195), at least for the u spinors. A similar calculation holds for v. The standard choice of phase is A = iy1 y3 = -iySC and Eq. (3-196) is finally established.
(3-197)
3-4-4 Summary
For the Dirac field we can now summarize the various transformation properties of quadratic forms under discrete symmetries. Define them according to their tensor character:
S(x) V!'(x) TIlV(x)
= : lj/(x)ljJ(x):
lj/(x)yl'ljJ(x):
= : lj/(x)al'vljJ(x):
(3-198)
AI'(x) = : lj/(x)ySyl'ljJ(x) : P(x) = i : lj/(x)ysljJ(x) :
QUANTIZATION-FREE FIELDS
The last factor i is chosen for hermiticity. The symbol A for the pseudovector (or "axial" vector) current (see below) is conventional and should, of course, not be confused with the vector potential. Using the results of the previous subsections and keeping in mind that the field anticommute and that :T is antiunitary, we have the following table:
S(x) f!J> S(X)
P(x)
~(x)
ptV(x) Tllv(x) - ptV(x) - TIl .( -x) ptV( -x)
AIl(X) -AIl(x) AIl(X) AIl(-x) -AIl( -x)
P(x) -P(x) P(x) -P(-x) P(-x)
S(x)
- P(x)
(3-199)
:T S(-X)
VIl ( -x)
- P( -x)
S(-x)
In this table each entry represents the action of the operator on the corresponding density. For instance, :TptV(x):Tt = - Tllv ( - x). We have added a line for the combined 0 = f!J>~:T antiunitary operation. The corresponding transformation laws of the electromagnetic vector potential A~m(x) are
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