Classical Dynamics in .NET framework

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12-1-2 Classical Dynamics
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Our aim is now to define a gauge-invariant action. As far as the coupling to the various multiplets of charged fields (matter fields) is concerned, it suffices to use the minimal coupling prescription. We substitute everywhere the covariant derivative for the ordinary one: (12-25) where AI' = Al'a ya is understood to act in the representation of the matter field. The part of the action depending on A only must be a Lorentz scalar, gaugeinvariant quantity-at most quadratic in the derivatives. The only candidate is the trace of Fl'vFI'V in some irreducible representation. For a simple Lie group traces in different irreducible representations are proportional since there exists a single quadratic invariant in the Lie algebra. We pick, for instance, the fundamental representation (of smallest dimension) and normalize its generators according to (12-26) where the minus sign is related to our convention that the tare antihermitian. For
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QUANTUM FIELD THEORY
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instance, in SU(2) and SU(3), we take respectively
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= 1,2,3
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and The action pertaining to A reads
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a = 1, ... ,8
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2~2 f d x tr (F"v PV) = - 4~2 f d x ~ F"vaPva
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(12-27)
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The dimensionless parameter g, not to be confused with a group element, plays the role of a coupling constant. This is evident after rescaling A into
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A-4gA
d x tr (F"vPV)
where
(12-28)
For the sake of brevity, however, we will not perform this rescaling until we derive the Feynman rules. For a general invariance group the Lie algebra is the direct sum of simple Lie algebras, plus abelian factors. To each of these terms a quadratic invariant and an independent coupling constant may be associated. An example is provided by the Weinberg-Salam model for the weak and electromagnetic interactions, based on the group SU(2) x U(l), with two coupling constants related to the Fermi coupling and the electric charge (see Sec. 12-6 below). The classical equations of motion are readily derived from the stationarity condition of the action (12-27):
0= M
:2 fd x tr [bAV(8"F"v - [A",F"v])]
Hence
[D", F"v] = 8"F"v - [A", F"v] = 0
(12-29)
(D")abF"vb = 0
which provides a nonabelian generalization of the Maxwell equations. The nonlinear character of the equations (12-29) makes their resolution nontrivial.
These equations possess the desired covariance property. If A. is a solution, so are its gauge transforms. Also, it must be clear that the system (12-29) is a compatible system. In particular, the
NONABELIAN GAUGE FIELDS
contraction with
a gives zero:
aVa"F"v
= = = =
aV[A", F"v] [aVA",F"v] + [A",avF"v] [O"A",F"v] + [A", [Av,F"v]] , 1[0"A" - a" AV- [A V A"], F"v]
Setting g (1-105)]
= 1 the canonical energy momentum tensor is [compare with Eq.
01'V
2 tr (FI'PoVAp - !gI'VFPUFpu )
but is not gauge invariant. This may be cured by subtraction of a total derivative L\0I'v:
where use has been made of the equation of motion (12-29): el'v = 01'v - L\01'v = 2 tr (FI'PPp - !gI'VFPUFpu ) = L (n pFp~ - !gl'V Ft:uFupa)
(12-30)
We introduce the analogs of the electric and magnetic fields
E~ =
FiO a
i, j, k
1, 2, 3
(12-31)
where, as throughout this chapter, the indices i, j, k are space indices (i, j, k = 1, 2, 3), while a, b, c are indices of the Lie algebra. In terms of E and B,
"2 ~ (Ea Ea + Ba Ba) =
tr (E2
+ B2)
(12-32)
e Oi = L (Ea
x Ba)i = - 2 tr (E x B)i
12-1-3 Euclidean Solutions to the Classical Equations of Motion
The search for classical solutions is motivated by the belief that a semiclassical approach may shed some light on the underlying quantum world and that classical configurations of fields that make the action stationary play an important role. Especially interesting are nondissipative configurations with a finite energy, i.e., such that their energy remains localized in a finite spatial region and is not radiated to infinity. Such objects are candidates to describe extended systems at the quantum level. These are coherent states of the fundamental fields, provided they are stable against decay. Stability may follow from some conservation law, perhaps of topological nature. These systems have received the name of solitons or energy lumps. As they arise from an expansion about a nontrivial stationary point of the action, these lumps and their quantum excitations exhibit features that could not have been suspected from the ordinary perturbative expansion. We will examine in Sec. 12-5-3 an example of a four-dimensional gauge theory involving scalar fields, which possesses finite energy solutions. On the other hand, it is possible to show that nontrivial finite-energy non dissipative solutions do not exist in nonabelian gauge theories with gauge fields only. In other words, any solution of this sort is equivalent to A" == O.
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