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W (q, p) d p = | (q)|2 ,
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36 The Feynman Path Integral and Coherent States
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The path integral was introduced by Feynman in 1948 as an alternative formulation of (nonrelativistic) quantum mechanics [27] Starting from the Schr dinger equation (in which and in the sequel we put = 1) i (x, t) 1 2 (x, t) + V (x) (x, t) = t 2m x 2 (390)
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for a particle of mass m moving in a potential V (x), a solution can be written as an integral, (x, t) = K (x, t; x , t ) (x , t ) dx , (391)
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36 The Feynman Path Integral and Coherent States
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which represents the wave function (x, t) at time t as a linear superposition over the wave function (x , t ) at the initial time t , t < t The integral kernel or propagator K (x, t; x , t ) can be formally expressed as an integral running over all continuous paths x (u), t u u u t , where x (t) = x and x (t ) = x are xed end points for all paths K (x, t; x , t ) = N e i [(m/2) x (u) V (x (u))] du Dx
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(392)
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Note that the integrand involves the classical Lagrangian for the system To give some meaning to this mathematically ill-de ned object, Feynman adopted the following lattice regularization with spacing : K (x, t; x , t ) = lim (m/2 i )(N +1)/2
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(m/2 )(x l+1 x l ) V (x l )
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dx l , (393)
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where x N +1 = x, x 0 = x , and = (t t )/(N + 1), N {1, 2, 3, } This procedure that yields well-de ned integrals has to be validated by the existence of a continuum limit as 0 Following the original Feynman approach, various authors, such as Feynman himself, Kac, Gel fand, Yaglom, Cameron, and It , attempted to nd a suitable continuous-time regularization procedure along with a subsequent limit to remove that regularization that ultimately should yield the correct propagator (see the illuminating review by Klauder [28], from which a large part of this section is borrowed) Now, it appears that a phase space formulation of path integrals is more natural, as was also suggested by Feynman [29], and later successfully carried further by Daubechies and Klauder [2, 30] Feynman (1951) proposed for the propagator the following integral on paths in the phase space K (q, t; q , t ) = M exp i p q H(q , p ) du D p Dq (394)
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Here one integrates over all paths q (u), t u u u t, with q (t) = q and q (t ) = q held xed, as well as over all paths p (u), t u u u t, without restriction A lattice space version of this expression is commonly given by
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H( p l+1/2 , 1 (q l+1 + q l )) 2
(395)
Like before = (t t )/(N + 1) The integration is performed over all p and q variables except for q N +1 = q and q 0 = q It is important to observe the presence of restrictions due to the canonical formalism of quantum mechanics: since q l implies
3 The Standard Coherent States: the (Elementary) Mathematics
a sharp q value at time t + l , the conjugate variable has been denoted by p l+1/2 to emphasize that a sharp p value must occur at a different time, here at t + (l + 1/2) , since it is not possible to have sharp p and q values at the same time Note that there is one more p integration than q integration in this formulation This discrepancy becomes clear when one imposes the composition law that requires that K (q, t; q , t ) = K (q, t; q , t ) K (q , t ; q , t ) dq , (396)
a relation that implies, just on dimensional grounds, that there must be one more p integration than q integration in the de nition of each K expression The contribution of Daubechies and Klauder (1985) was to reexamine (394) by using a complete phase space formalism combined with coherent states through the Fock Bargmann representation of wave functions and operators They introduced coherent states, denoted here by |q, p and de ned as