16.3. Representation of quantum amplitudes
167
state of lowest energy E 0 (the ground state) provides the dominant contribution. Thus, in this limit, our amplitude will represent the process in which
the system begins in its ground state |Ω› at τ i → −∞, with q = q
i , and ends
in |Ω› at τ f → ∞, with q = q
f .
How do we represent propagators in this formalism? Consider the expression (somewhat analogous to a field theory propagator)
G fi (t a , t
f
b ) ≡ ‹q t f |T {q ˆ H (t a )ˆ q
i
H (t b )} |q ti › ,
(16.79)
where T is the usual time-ordering operator. Using (16.51) and (16.52),
(16.79) can be written, for t b > t a , as
ˆ
ˆ
ˆ
G (t , t ) = ‹q
f
|e
−iH(t f −t b ) q ˆe
−iH(t b −ta)
H
a
q ˆe
−i (ta
fi
b
−ti)
|q
i
› .
(16.80)
Inserting a complete set of states and Euclideanizing, (16.80) becomes
∫
ˆ
G fi (t a , t b ) =
dq
a dq
b q
a q
b
‹q
f
|e
−H(τ f −τ b )
|q
b
›
× ‹q
b
|e
− ˆ
ˆ
H(τ b −τa)
|q
a
›‹q
a
|e
−H(τa−τi)
|q
i
› .
(16.81)
Now, each of the three matrix elements has a discretized representation of the
form (16.63), with say N 1 − 1 variables in the interval (τ a , τ i ), N 2 − 1 in (τ b , τ a )
and N 3 − 1 in (τ f , τ b ). Each such representation carries one ‘surplus’ factor
of [A(∈)]
−1 , making an overall factor of [A(∈)]
−3 . Two of these factors can be
associated with the dq
a dq
b integration in (16.81), so that we have a total of
N 1 + N 2 + N 3 − 1 properly normalized integrations, and one ‘surplus’ factor
[A(∈)]
−1 as in (16.66). If we now identify q(τ a ) ≡ q
a , q(τ n ) ≡ q
b , it follows
that (16.81) is simply
∫
τ
D
∫ f L dτ
q(τ )q(τ a )q(τ b )e
− τ i
.
(16.82)
In obtaining (16.82), we took the case τ b > τ a . Suppose alternatively that
τ a > τ b . Then the order of τ a and τ b inside the interval (τ i , τ f ) is simply
reversed, but since q
a and q
b in (16.81), or q(τ a ) and q(τ b ) in (16.82), are
ordinary (commuting) numbers, the formula (16.82) is unaltered, and actually
does represent the matrix element (16.79) of the time-ordered product.
16.3.2 Quantum field theory
The generalizations of these results to the field theory case are intuitively
clear. For example, in the case of a single scalar field φ(x), we expect the
analogue of (16.82) to be (cf (16.4))
∫
[ ∫ τ f
]
Dφ(x) φ(x a )φ(x b )exp −
τi
L E (φ, ∇φ, ∂ τ φ) d
4 x E ,
(16.83)
167
state of lowest energy E 0 (the ground state) provides the dominant contribution. Thus, in this limit, our amplitude will represent the process in which
the system begins in its ground state |Ω› at τ i → −∞, with q = q
i , and ends
in |Ω› at τ f → ∞, with q = q
f .
How do we represent propagators in this formalism? Consider the expression (somewhat analogous to a field theory propagator)
G fi (t a , t
f
b ) ≡ ‹q t f |T {q ˆ H (t a )ˆ q
i
H (t b )} |q ti › ,
(16.79)
where T is the usual time-ordering operator. Using (16.51) and (16.52),
(16.79) can be written, for t b > t a , as
ˆ
ˆ
ˆ
G (t , t ) = ‹q
f
|e
−iH(t f −t b ) q ˆe
−iH(t b −ta)
H
a
q ˆe
−i (ta
fi
b
−ti)
|q
i
› .
(16.80)
Inserting a complete set of states and Euclideanizing, (16.80) becomes
∫
ˆ
G fi (t a , t b ) =
dq
a dq
b q
a q
b
‹q
f
|e
−H(τ f −τ b )
|q
b
›
× ‹q
b
|e
− ˆ
ˆ
H(τ b −τa)
|q
a
›‹q
a
|e
−H(τa−τi)
|q
i
› .
(16.81)
Now, each of the three matrix elements has a discretized representation of the
form (16.63), with say N 1 − 1 variables in the interval (τ a , τ i ), N 2 − 1 in (τ b , τ a )
and N 3 − 1 in (τ f , τ b ). Each such representation carries one ‘surplus’ factor
of [A(∈)]
−1 , making an overall factor of [A(∈)]
−3 . Two of these factors can be
associated with the dq
a dq
b integration in (16.81), so that we have a total of
N 1 + N 2 + N 3 − 1 properly normalized integrations, and one ‘surplus’ factor
[A(∈)]
−1 as in (16.66). If we now identify q(τ a ) ≡ q
a , q(τ n ) ≡ q
b , it follows
that (16.81) is simply
∫
τ
D
∫ f L dτ
q(τ )q(τ a )q(τ b )e
− τ i
.
(16.82)
In obtaining (16.82), we took the case τ b > τ a . Suppose alternatively that
τ a > τ b . Then the order of τ a and τ b inside the interval (τ i , τ f ) is simply
reversed, but since q
a and q
b in (16.81), or q(τ a ) and q(τ b ) in (16.82), are
ordinary (commuting) numbers, the formula (16.82) is unaltered, and actually
does represent the matrix element (16.79) of the time-ordered product.
16.3.2 Quantum field theory
The generalizations of these results to the field theory case are intuitively
clear. For example, in the case of a single scalar field φ(x), we expect the
analogue of (16.82) to be (cf (16.4))
∫
[ ∫ τ f
]
Dφ(x) φ(x a )φ(x b )exp −
τi
L E (φ, ∇φ, ∂ τ φ) d
4 x E ,
(16.83)