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Parity in VS .NET
341 Parity Read PDF417 2d Barcode In Visual Studio .NET Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in VS .NET applications. PDF 417 Encoder In .NET Framework Using Barcode generator for VS .NET Control to generate, create PDF417 2d barcode image in Visual Studio .NET applications. 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 (3177) 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 (3177) entails Recognize PDF417 In .NET Using Barcode scanner for .NET framework Control to read, scan read, scan image in .NET framework applications. Bar Code Encoder In VS .NET Using Barcode encoder for Visual Studio .NET Control to generate, create bar code image in .NET framework applications. f!J>bAp)f!J>t = I'/pbAv) y0u<a)(V) = u(a)(p) Bar Code Scanner In VS .NET Using Barcode reader for VS .NET Control to read, scan read, scan image in .NET applications. Drawing PDF417 In Visual C# Using Barcode drawer for .NET Control to generate, create PDF 417 image in .NET applications. (3178) PDF417 2d Barcode Encoder In .NET Framework Using Barcode generation for ASP.NET Control to generate, create PDF417 2d barcode image in ASP.NET applications. PDF 417 Printer In Visual Basic .NET Using Barcode drawer for VS .NET Control to generate, create PDF417 image in .NET framework applications. QUANTUM FIELD THEORY
UCC  12 Drawer In .NET Framework Using Barcode generation for .NET Control to generate, create GS1128 image in .NET applications. Generate Matrix 2D Barcode In VS .NET Using Barcode printer for .NET Control to generate, create Matrix Barcode image in Visual Studio .NET applications. Irrespective of the phase rfp, the relative parity of a fermionantifermion system is minus one and q>~ is a multiple of the identity. A unitary operator fulfilling the above requirements is Encoding GS1 RSS In VS .NET Using Barcode generation for Visual Studio .NET Control to generate, create GS1 RSS image in VS .NET applications. USS Codabar Generator In Visual Studio .NET Using Barcode encoder for Visual Studio .NET Control to generate, create ABC Codabar image in .NET framework applications. q> = exp
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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 (3180) Show that the bilinear Dirac current has a similar behavior: (3181) from which follows, using d4 x
d4 5i;, that the interaction lagrangian
is parity invariant.
342 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) (3182) for reasons to become clear below.
QUANTIZATIONFREE FIELDS
From Chap. 2 the corresponding action on a Dirac field is
~I/!(x)~t = YfeCfjJT(X) (3183) 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, (3184a) 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 ~) (3184b) In the usual representation X (k) u kX (k) = =+= X (k). Furthermore, iu 2 ({J l (k), and we can verify that
(3185) CvT(k, ) = u(k, ) With these choices we readily derive from (3183) that
~b(k, )~t =
Yfed(k, ~dt(k, )~t =
Yfebt(k, ) (3186) This could have been imposed at first, with (3183) 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
(3187) A[b t(k, e)b(k, e)  dt(k, e)d(k, e)] (3188) [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 fermionantifermion 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) (3189) 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 FermiDirac statistics. The positronium states are unstable and have a slow decay by photon emission. From (3182) the electromagnetic potential is odd under 'C. This is, in fact, a condition for electromagnetic interactions to be invariant under 'C. Hence an nphoton 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 =

