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Among the striking consequences emerging when combining microcausality with relativistic invariance, let us quote the spin statistics relation (half-integer spin particles are fermions, integer spin ones are bosons) and the existence of a TCP invariance. The latter involves a product of time reversal (T), parity (P), and charge conjugation (C). This implies the existence of antiparticles with the same kinematic invariants as their particle counterparts and opposite additive quantum numbers (electric, baryonic, leptonic charges, etc.).
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3-1-1 General Formulation
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Let CPa(x) be the fields whose dynamics we propose to study. The index Ii. stands for internal characteristics (charges, etc.) or kinematic ones (such as Lorentz indices). For the moment we assume these fields to be free of constraints (which would reduce the number of degrees of freedom). We provisionally set aside half-integer spin fields, the correct treatment of which requires special considerations (Sec. 3-3). From the classical Lagrange function at a fixed time
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we derive the conjugate fields
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d 3 x 2(cp, ocp)
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(3-1)
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(3-2) To construct the hamiltonian operator H we first replace the c-number fields by operators satisfying canonical equal-time commutation rules: (3-3) with the commutators [cp, cp] and [n, n] vanishing. After inverting (3-2) to give ooCP in terms of nand cp we obtain Has
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[~na(t' x)OOCPa(t, x) -
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2(cp, ocp) ]
(3-4)
This procedure suffers from the usual drawbacks arising from the operator ordering. Moreover, the multiplication of operator fields at the same point will lead to new difficulties, as we shall soon realize. The two aspects are related. It must be stressed that when writing (3-3) we have not yet specified in which Hilbert space these operators act. This question has a simple answer in the case of free fields, as we shall see, and hence is bypassed when studying small perturbations around this situation. It is, however, entirely nontrivial in the general case, where its answer requires some knowledge of the dynamics. This is why the latter has some bearing on the very construction of the theory. To be specific, let us assume that we deal with only one real field cp in the
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classical picture-hence an hermitian cp in the quantum one-with a lagrangian
= ~(acp)2
- V(cp)
(3-5)
where V is a smooth function (a polynomial, say). The classical equations of motion read
(3-6)
If V reduces to a quadratic term V(cp) = (m 2/2)cp2,
! (cp)
= ~(acp
~m2cp2
(3-7)
from which follows the Klein-Gordon equation
(3-8)
to be interpreted here as a classical field equation and not as a relativistic generalization of Schrodinger's equation. For an arbitrary V(cp), that is, for an arbitrary self-interaction without derivatives, the conjugate momentum n is given by
(3-9)
so that the hamiltonian, expressed in terms of cp and n, is
and in the simple case of a quadratic V( cp) given by (3-7)
d 3x
{Hn 2 + (V~ ] + V(cp)}
(3-10)
d3 x
Hn 2 + (Vcp)2 + m2cp2]
(3-11)
We have here no problem of operator ordering. Hence the only source of trouble may come from multiplying operators at the same point. Let us stick for the time being to the case (3-11). We recognize a simple structure of coupled harmonic oscillators. In order to uncover its physical interpretation, let us pretend that space has only one dimension and that instead of assuming continuous values the coordinate x can take only discrete ones which are integral multiples of an elementary length taken as unity. In this case (3-11) would be replaced by
[n; + (CPn - CPn-l + m2 cp;]
(3-12)
A physical model which could be described by (3-12) would be the vibrations of a one-dimensional "crystal," with CPn standing for the displacement of the nth
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atom and nn the conjugate variable. Each individual oscillator with restoring force provided by the m 2 cp; term would be coupled to its nearest neighbors through the (CPn - CPn d contributions to the potential energy. To study such a model it is natural to search for the proper modes. Using the discrete translational invariance of H this leads us to introduce Fourier transforms in the form
nn iit(k)
f+1t dk eikn n -(k) 2
ii( -k)
(3-13)
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