QUANTUM FIELD THEORY in VS .NET

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512 QUANTUM FIELD THEORY
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11-1-2 Behavior of the Ground State
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The symmetry properties of a system may be characterized by the behavior of its ground state. As we shall see, if the vacuum is annihilated by a set of charges, the corresponding currents are conserved and the symmetry group unitarily implemented. Conversely, assuming conserved currents, if the vacuum is not invariant the symmetry is spontaneously broken. Were the vacuum not unique, we could define an orthonormal basis in this lowest-energy subspace, diagonalizing all commuting hermitian observables. To simplify, assume a discrete set of such states In). Consider any couple of observables A(x), B(y). When their arguments are infinitely space-like separated, only the intermediate translationally invariant ground states should contribute to the matrix elements of their product, according to a generalization of the RiemannLebesgue lemma lim
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<nl A(x)B(O) 1m) = I <nl A(O) Ip)<pl B(O) 1m)
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Here Ip) runs over the ground states. According to causality, the matrix element ofthe commutator <nl [A (x), B(O)] vanishes in the limit Ix 00. The matrices <nl A(O) and <nl B(O) commute and can be simultaneously diagonalized. Within a given sector the vacuum matrix element of a product of local operators factorizes when their space-like separations become large. This is called the cluster property. An apparent degeneracy of the vacuum can occur as a result of a poor approximation. The real ground state is the unique combination which minimizes the energy. An example is provided by a quantum mechanical one-dimensional system with a potential V(x) = (x 2 - I with two equal minima. Quantum mechanics allows for barrier tunneling and leads to a unique symmetric ground state. We may approach the ground state in a translationally invariant way by requiring it to minimize the effective potential Yeff(4)). The latter plays the role of density of potential energy in a state with a given mean value of the field.
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Let us quote an important property which follows from locality. Assume that a conserved current operator jP(x) has been constructed, so that opj"(x) = O. Define the integral of /(x, t) over a bounded region V of space, Qv(t), as
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(11-9)
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This operator clearly stands a better chance of being well defined than its limit Q(t), the integral over all space. The commutator of Qv(t) with a local observable A is independent of time for V large enough: . d hm - [Qv(t), A] = 0 (11-10) v~oo dt Indeed, from current conservation we have
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d3x [opj"(x, t), A]
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r d3
x [/(x, t), A]
r Js dS [j(x, t), A]
SYMMETRIES
When V becomes large enough, the surface integral vanishes since the commutator involves local operators separated by a very large space-like interval. The validity of this statement extends to more general situations. In particular, A may be replaced by any multilocal operator. The proof could even be extended to a nonrelativistic system provided that only short-range forces are present.
What seems an utter triviality-either the vacuum is invariant or it is notcovers, however, a wealth of situations with very different physical content. What ifthe vacuum is indeed invariant A theorem of Coleman shows that the associated curren ts are then conserved. Assume that Q(t), the space in tegral of/, is well defined (at least on a dense subset of the Hilbert space including the ground state) and annihilates the vacuum
Q(t)[O) = fd3XjO(X,t)[0) = 0
(11-11)
If the theory admits an energy gap, any state with zero momentum will be such that Pn = OUo(x, t) [0) = 0
from translational in variance, since the above quantity is x independent with a vanishing integral. Here we cheat slightly since strictly speaking such a state is not normalizable. The necessary refinement may, however, be worked out. Consequently,
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