p e i bracketleftBig p \u02c6 2 2 m V x \u02c6 j t j bracketrightBig \u03b4t x j 11 e iV x j t

P e i bracketleftbig p ˆ 2 2 m v x ˆ j t j

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)( p | e i bracketleftBig p ˆ 2 2 m + V ( x ˆ j ,t j ) bracketrightBig δt | x j ) (11) = e iV ( x j ,t j ) δt integraldisplay dp 2 π e i p 2 2 m δt e ip ( x j +1 x j ) (12) Now we can use the Gaussian integral in Eq. (6), integraltext dp exp ( 1 2 ap 2 + Jp ) = 2 π a radicalBig exp parenleftBig J 2 2 a parenrightBig , with a = i δt m and J = i ( x j +1 x j ) to get ( x j +1 | e iHδt | x j ) = Ne iV ( x j ,t j ) δt e i m 2 δt ( x j +1 - x j ) 2 ( δt ) 2 = Ne iL ( x,x ˙) δt (13) where N is an x - and t -independent normalization constant, which we’ll justify ignoring later, and L ( x,x ˙) = 1 2 mx ˙ 2 V ( x,t ) (14) is the Lagrangian. We see that the Gaussian integral performed the Legendre transform to go from H ( x,p ) to L ( x,x ˙) . Using Eq. (13), each term in Eq. (10) becomes just a number and the product reduces to ( f | i ) = N n integraldisplay dx n dx 1 e iL ( x n ,x ˙ n ) δt e iL ( x 1 ,x ˙ 1 ) δt (15) Finally, taking the limit δt 0 the exponentials combine in to an integral over dt and we get ( f | i ) = N integraldisplay x ( t i )= x i x ( t f )= x f D x ( t ) e iS [ x ] (16) 4 Section 2
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where D x means sum over all paths x ( t ) with the correct boundary conditions and the action is S [ x ] = integraltext dtL [ x ( t ) ,x ˙( t )] . Note that N has been redefined and is now formally infinite, but it will drop out of any physical quantities, as we will see in the path integral case. 2.3 Path integral in quantum field theory The field theory derivation is very similar, but the set of intermediate states is more compli- cated. We’ll start by calculating the vacuum matrix element ( 0; t f | 0; t i ) . In quantum mechanics we broke the amplitude down into integrals over | x )( x | for intermediate times where the states | x ) are eigenstates of the x ˆ operator. In field theory, the equivalent of x ˆ are the Schrödinger picture fields φ ˆ ( x ) , which at any time t can be written as φ ˆ ( x ) = integraldisplay d 3 p (2 π ) 3 1 2 ω p radicalbig ( a p e ipx + a p e ipx ) (17) Each field comprises an infinite number of operators, one at each point x . We put the hat on φ to remind you that it’s an operator. Up to this point, we have been treating the Hamiltonian and Lagrangian as functionals of fields and their derivatives. Technically, the Hamiltonian should not have time derivatives in it, since it is supposed to be generating time translation. Instead of t φ the Hamiltonian should depend on canonical conjugate operators which we introduced in Lecture I-2 as π ˆ( x ) ≡ − i integraldisplay d 3 p (2 π ) 3 ω p 2 radicalBig ( a p e ipx a p e ipx ) (18) and satisfy bracketleftbig φ ˆ ( x ) ˆ( y ) bracketrightbig = 3 ( x y ) (19) These canonical commutation relations and the Hamiltonian which generates time-translation define the quantum theory. The equivalent of | x ) is a complete set of eigenstates of φ ˆ φ ˆ ( x ) | Φ ) = Φ( x ) | Φ ) (20) The eigenvalues are functions of space Φ( x ) . 1 The equivalent of | p ) are the eigenstates of π ˆ( x ) which satisfy π ˆ( x ) | Π ) = Π( x ) | Π ) (21) The | Π ) states are conjugate to the | Φ ) states, and satisfy ( Π | Φ ) = exp parenleftbigg i integraldisplay d 3 x Π( x ) Φ( x ) parenrightbigg (22)
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