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Unformatted text preview: 5 Path Integrals in Quantum Mechanics and Quan- tum Field Theory In the past chapter we gave a summary of the Hilbert space picture of Quantum Mechanics and of a Scalar Quantum Field Theory. Here we will present the Path Integral picture of Quantum Mechanics ans Scalar Quantum Field Theory. The Path Integral picture is important for two reasons. First, it offers an alternative, complementary, picture of Quantum Mechanics in which the role of the classical limit is apparent. Secondly, it gives a direct route to the study regimes where perturbation theory is either inadequate or fails completely. A standard approach to these problems is the WKB approximation (Wentzel, Kramers and Brillouin). As it happens, it is extremely difficult (if not im- possible) the generalize the WKB approximation to a Quantum Field Theory. Instead, the non-perturbative treatment of the Feynman path integral, which is equivalent to WKB, is generalizable to non-perturbative problems in QFT. In this chapter we will use path integrals only for bosonic systems, such as scalar and abelian gauge fields. In the following chapters we will also give a full treatment of the path integral, including its applications to fermionic fields, non-relativistic many body systems, and Abelian gauge fields. Non-Abelian gauge fields will be discussed in the following course, Physics 583. 5.1 Path Integrals and Quantum Mechanics Consider a simple quantum mechanical system whose dynamics can be described by a (generalized) time-dependent coordinate operator q ( t ), i.e., the position operator in the Heisenberg representation. We will denote by | q,t ) an eigenstate of q ( t ) with eigenvalue q , q ( t ) | q,t ) = q | q,t ) (1) We want to compute the amplitude F ( q f ,t f | q i ,t i ) = ( q f ,t f | q i ,t i ) (2) Let q S be the Schrodinger operator, related to the Heisenberg operator q ( t ) by the action of the time evolution operator: q ( t ) = e i Ht/ planckover2pi1 q S e i Ht/ planckover2pi1 (3) and whose eigenstates are | q ) , q S | q ) = q | q ) (4) The states | q ) and | q,t ) are related via the evolution operator | q ) = e i Ht/ planckover2pi1 | q,t ) (5) Therefore the amplitude F ( q f ,t f | q i ,t i ) is a matrix element of the evolution operator F ( q f ,t f | q i ,t i ) = ( q f | e i H ( t i t f ) / planckover2pi1 | q i ) (6) 1 The amplitude F ( q f ,t f | q i ,t i ) has a simple physical interpretation. Let us set, for simplicity, | q i ,t i ) = | , ) and | q f ,t f ) = | q,t ) . Then, from the definition of this matrix element, we find out that it obeys lim t F ( q,t | , 0) = ( q | ) = ( q ) (7) Furthermore, after some algebra we also find that i planckover2pi1 F t = i planckover2pi1 t ( q,t | , ) = i planckover2pi1 t ( q | e i Ht/ planckover2pi1 | ) = ( q | He i Ht/ planckover2pi1 | ) = integraldisplay dq ( q | H | q )( q | e i Ht/ planckover2pi1 | ) (8) where we have used that, since...
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