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The path integral formalism of Feynman and Kac provides a unified view of quantum mechanics, field theory, and statistical models. Starting from the case of finitely many degiees offreedom it is generalized to include fermionic systems and then extended to infinite systems. The steepest-descent method of integration exhibits the close relationship with classical mechanics and allows us to recover ordinary perturbation theory. Among various applications, we deal here with the concept of effective action, quantization of constrained systems, and evaluation of high orders in perturbation theory.
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9-1 PATH INTEGRALS
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The original suggestion of an alternative presentation of quantum mechanical amplitudes in terms of path integrals stems from the work of Dirac (1933) and was brilliantly elaborated by Feynman in the 1940s. Schwinger developed an equivalent approach based on functional differentiation. This work was first regarded with some suspicion due to the difficult mathematics required to give it a decent status. In the 1970s it has, however, proved to be the most flexible tool in suggesting new developments in field theory and therefore deserves a thorough presentation.
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9-1-1 The Role of the Classical Action in Quantum Mechanics
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Let us return to quantum mechanics to inquire about the role of the lagrangian formalism as opposed to the hamiltonian one. For simplicity let us discuss a
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QUANTUM FIELD THEORY
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system with one degree of freedom described by the pair of conjugate operators Q and P satisfying
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[Q,P] = i
(9-1)
We use capital letters for operators to distinguish them from their classical cnumber counterparts. Let the hamiltonian be
p2 H(P, Q) = 2m
+ V(Q)
(9-2)
and denote by la), Ib), ... the states of the system. We are going to seek an expression for the transition amplitude
<b(t') la(t = <bl
e-iH(t'-t)
(9-3)
With the usual representation of the commutation rules (9-1) we could introduce the square integrable wave functions
a(q)
<q Ia)
a(p) = <p Ia)
and attempt to solve the Schrodinger partial differential equation arising from (9-2). Here the improper states Iq) and Ip) are such that
Qlq)=qlq) <q' Iq)
plp)=plp) <qlp) = <plq)* = _l_e ipq (9-4)
t5(q' - q)
<p'lp) = t5(p' - p)
<qlplp) = p<qlp) = -=- -;- <qlp) I uq
Returning to the transition amplitude, we note that we may use the superposition principle to insert a complete set of states at intermediate times. This is analogous to the use of the Huyghens' principle in optics. We split the time evolution in infinitesimal steps t --+ t + /::;.t and first evaluate
<q2(t
+ /::;.t)lql(t =
<q21
idtH
Iql)
The boundary condition requires that when /::;.t goes to zero the above amplitude reduces to t5(q2 - qd. For smaIl /::;.t we expect that the matrix element is negligible when q2 differs appreciably from q!, whether its modulus decreases or its phase oscillates very rapidly. This suggests that we may substitute for the potential operator V(Q) its value V(qd or V(q2), and amounts to an approximation of the type
The terms we have neglected involve the commutators [p2/2m, V(Q)] multiplying higher powers of flt. They will be negligible if V has a slow variation in the neighborhood of ql and q2. This means that in the short interval /::;.t we neglect
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the transfer of potential into kinetic energy. For the matrix element we obtain the estimate:
(9-5) We observe the consistency of this procedure. For Iq2 - q11 (hM/m)1/2 the strong oscillations damp the amplitude so that the correction terms are vanishingly small provided IV'/V I(Mt/m)1/2 1, where V' is the derivative of the potential. This may be used to obtain a more symmetric form by replacing V(ql) by HV(q1) + V(q2)], for instance. Expression (9-5) is suggestive. Indeed, (q2 - q1)/M is the analog of the velocity 4, and the exponent reduces to iML(q, 4) where L(q, 4) is the Lagrange function (9-6) L(q, 4) = im4 2 - V(q)
(t, t
More precisely, let q(t') be the trajectory from q1 to q2 during the time interval + M) according to classical mechanics, i.e., obeying the principle of least action. As we have just seen, in the limit M -+ 0, only values of the kernel for q2 in the vicinity of q1 matter, in a domain of order (M/m)1/2. The phase of the transition amplitude is then equivalent to the action
1(2,1) =
qZ(t+M) dt' L(q, 4)
(9-7)
q l(t)
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