Parity in VS .NET

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3-4-1 Parity
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We assume that it is possible, with appropriate modification of the experimental apparatus, to obtain the parity transformed state. From the field point of view we look for a unitary operator f!J> satisfying (3-177) since from the previous chapter we know that yOl/J(x) satisfies the parity transformed Dirac equation. We thus expect f!J> to commute with H. Equation (3-177) entails
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f!J>bAp)f!J>t = I'/pbAv) y0u<a)(V) = u(a)(p)
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(3-178)
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QUANTUM FIELD THEORY
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Irrespective of the phase rfp, the relative parity of a fermion-antifermion system is minus one and q>~ is a multiple of the identity. A unitary operator fulfilling the above requirements is
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q> = exp
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dk "=~'2 {b!(k{Abik) + ~ b,,(k)] - dHk{(A + n)d,,(k) + ~ d,,(k)]}
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(3-179)
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where A is arbitrary and rfp = e- i(H1t/2). In particular, rfp = 1 if A = -nI2. A simple means of verifying that q> is indeed a symmetry is to observe that under a time translation
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Hence,
eiHa q> e- iHa = q>
We may further verify that ! l'P~! l't = P w In a coupled theory the complete parity operator will be the product of those pertaining to the different fields. Construct in particular ! l' A for the electromagnetic field such that
(3-180)
Show that the bilinear Dirac current has a similar behavior:
(3-181)
from which follows, using d4 x
d4 5i;, that the interaction lagrangian
is parity invariant.
3-4-2 Charge Conjugation
We have seen that quantized fields may describe particles of opposite charge with identical masses and spin. The corresponding charge may have a different interpretation according to the physical context. It can be electric, baryonic, leptonic, etc., whichever the case may be. Charge conjugation invariance therefore implies
1. The existence of antiparticles 2. The symmetric behavior of both kinds of quanta
We had a first example with the charged scalar field. It may, however, occur that particles and antiparticles are identical. Such is the case for photons, where the corresponding operator 'fl just reverses the sign of the field
'flAJl(x)'fl t = -AJl(x)
(3-182)
for reasons to become clear below.
QUANTIZATION-FREE FIELDS
From Chap. 2 the corresponding action on a Dirac field is
~I/!(x)~t = YfeCfjJT(X)
(3-183)
where transposition refers only to Dirac indices. In the standard representation of y matrices C = iyOy2 To be definite, we choose creation and annihilation operators for the helicity states b(k, ), d(k, ). Then
+ 1) =
J2m(kO
~ +m
+ m)
(((J (k)) . = bee,
(3-184a)
where and by definition
u k({J (k) =
({J (k)
((Jt(k)({Je,(k)
v(k, )=CuT(k, )=
J 2m(m + kO) (O(ki':\) -~+m X ~)
(3-184b)
In the usual representation X (k) u kX (k) = =+= X (k). Furthermore,
iu 2 ({J l (k), and we can verify that
(3-185)
CvT(k, ) = u(k, )
With these choices we readily derive from (3-183) that
~b(k,
)~t =
Yfed(k,
~dt(k, )~t =
Yfebt(k, )
(3-186)
This could have been imposed at first, with (3-183) following as a consequence. Up to a phase factor, ~ interchanges particles and antiparticles with the same momentum, energy, and helicity. The vacuum is left invariant. An explicit expression for ~ is
~'" = ~1~2
= exp -
~2 = exp i
f ~f
(3-187)
A[b t(k, e)b(k, e) - dt(k, e)d(k, e)]
(3-188)
[bt(k, e) - dt(k, e)] [b(k, e) - d(k, e)]
with Yfe = eiA The only effect of ~ 1 is to carry this phase, and Yfe = 1 corresponds to ~1 = 1. The reader will also check that the current: f//YI'I/!: is odd under charge conjugation.
As an application let us classify, according to charge conjugation, the lowest bound states of a fermion-antifermion system, the prototype of which is positronium. The latter is an (e+ e-) system analogous to the hydrogen atom (pe-) with the proton replaced by a positron. A nonrelativistic description is justified as a first approximation, due to the weakness
154 QUANTUM FIELD THEORY
of electromagnetic binding forces. The ground state is an s wave, n = 1, but hyperfine effects split a triplet orthopositronium state S!, if we use the notation 2S+ 1 L J ) from a singlet So) parapositronium state. Simplified wave functions correct from the quantum number point of view are written, using a fixed axis for spin quantization, as
IJ = 1, M = 0, ortho) = IJ = 0, M = 0, para) =
d 3 q 'Pl(1 q I) [b!. (q)dt( -q)
+ H(q)d!.( -q)] 10)
H(q)d!.( -q)] 10)
d 3 q 'Po(1
ql)[b!.(q)d~( -q) -
The relative momentum wave functions 'PI and 'Po only depend on the magnitude of q; H(q) [or d1o(q)] denotes an electron (positron)-creation operator of momentum q with spin i along a fixed axis. Charge conjugation reads
The arbitrary phase 1'/c disappears when 'C acts on these states with the result that 'C lortho) 'Clpara)
-Iortho) (3-189) Ipara)
The signs arise as follows. Charge conjugation interchanges electron and positron, as a result of which the relative momentum changes sign, leading to a factor of (_l)L = 1 for s waves; the spin indices are interchanged, leading to a plus (minus) sign for a triplet (singlet) state. Finally, there is an additional minus sign arising from the anticommutation of b t and d t operators. This is an indirect and unexpected manifestation of Fermi-Dirac statistics. The positronium states are unstable and have a slow decay by photon emission. From (3-182) the electromagnetic potential is odd under 'C. This is, in fact, a condition for electromagnetic interactions to be invariant under 'C. Hence an n-photon state behaves as
Correspondingly, orthopositronium must decay into an odd number of photons, and parapositronium into an even number. One photon decay is forbidden for the ortho state by energy momentum conservation. rt must decay into at least three photons, while parapositronium can decay into two photons and has therefore a much shorter lifetime. The coupling constant being the fine structure constant a, for the lifetime, we expect the ratio 'singlet/'triplet ~ O(a). We shall compute these quantities in Chap. 5. To lowest order in a, rs =
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