162
16. Lattice Field Theory, and the Renormalization Group Revisited
the a → 0 limit of the quantum theory is, as we shall see in section 16.5,
more subtle than the naive replacements (16.5) because of renormalization
issues, as should be no surprise to the reader by now). However, we have
not yet considered how we are going to turn this classical lattice theory into
a quantum one. The fact that the calculations are mostly going to have to
be done numerically seems at once to require a formulation that avoids noncommuting operators. This is precisely what is provided by Feynman’s sum
over paths formulation of quantum mechanics and of quantum field theory,
and it is therefore an essential element in the lattice approach to quantum
field theory. In this section we give a brief introduction to this formalism,
starting with quantum mechanics.
16.3.1 Quantum mechanics
In section 5.2.2 we stated that in this approach the amplitude for a quantum
system, described by a Lagrangian L depending on one degree of freedom q(t),
to pass from a state in which q = q
i at t = t i to a state in which q = q
f at
time t = t f , is proportional to (with ħ = 1)
( ∫
∑
t f
)
exp i
L(q(t), q˙(t))dt ,
(16.49)
t
all paths q(t)
i
where q(t i ) = q
i , and q(t f ) = q
f . We shall now provide some justification for
this assertion.
We begin by recalling how, in ordinary quantum mechanics, state vectors
and observables are related in the Schr¨ odinger and Heisenberg pictures (see
appendix I of volume 1). Let q ˆ be the canonical coordinate operator in the
Schr¨ odinger picture, with an associated complete set of eigenvectors |q› such
that
q ˆ|q› = q|q› .
(16.50)
The corresponding Heisenberg operator q ˆ H (t)is defined by
ˆ
q ˆ H (t) = e
iH(t−
ˆ
t0) q ˆe
−iH(t−t0)
(16.51)
ˆ
where H is the Hamiltonian, and t 0 is the (arbitrary) time at which the two
pictures coincide. Now define the Heisenberg picture state |q t › H by
|q t ›
ˆ
= e
iH(t−t0)
H
|q› .
(16.52)
We then easily obtain from (16.50)–(16.52) the result
q ˆ H (t)|q t › H = q|q t › H ,
(16.53)
which shows that |q t › H is the (Heisenberg picture) state which at time t is an
eigenstate of q ˆ H (t) with eigenvalue q. Consider now the quantity
H ‹q
f
t f
|q
i
ti › H
(16.54)
16. Lattice Field Theory, and the Renormalization Group Revisited
the a → 0 limit of the quantum theory is, as we shall see in section 16.5,
more subtle than the naive replacements (16.5) because of renormalization
issues, as should be no surprise to the reader by now). However, we have
not yet considered how we are going to turn this classical lattice theory into
a quantum one. The fact that the calculations are mostly going to have to
be done numerically seems at once to require a formulation that avoids noncommuting operators. This is precisely what is provided by Feynman’s sum
over paths formulation of quantum mechanics and of quantum field theory,
and it is therefore an essential element in the lattice approach to quantum
field theory. In this section we give a brief introduction to this formalism,
starting with quantum mechanics.
16.3.1 Quantum mechanics
In section 5.2.2 we stated that in this approach the amplitude for a quantum
system, described by a Lagrangian L depending on one degree of freedom q(t),
to pass from a state in which q = q
i at t = t i to a state in which q = q
f at
time t = t f , is proportional to (with ħ = 1)
( ∫
∑
t f
)
exp i
L(q(t), q˙(t))dt ,
(16.49)
t
all paths q(t)
i
where q(t i ) = q
i , and q(t f ) = q
f . We shall now provide some justification for
this assertion.
We begin by recalling how, in ordinary quantum mechanics, state vectors
and observables are related in the Schr¨ odinger and Heisenberg pictures (see
appendix I of volume 1). Let q ˆ be the canonical coordinate operator in the
Schr¨ odinger picture, with an associated complete set of eigenvectors |q› such
that
q ˆ|q› = q|q› .
(16.50)
The corresponding Heisenberg operator q ˆ H (t)is defined by
ˆ
q ˆ H (t) = e
iH(t−
ˆ
t0) q ˆe
−iH(t−t0)
(16.51)
ˆ
where H is the Hamiltonian, and t 0 is the (arbitrary) time at which the two
pictures coincide. Now define the Heisenberg picture state |q t › H by
|q t ›
ˆ
= e
iH(t−t0)
H
|q› .
(16.52)
We then easily obtain from (16.50)–(16.52) the result
q ˆ H (t)|q t › H = q|q t › H ,
(16.53)
which shows that |q t › H is the (Heisenberg picture) state which at time t is an
eigenstate of q ˆ H (t) with eigenvalue q. Consider now the quantity
H ‹q
f
t f
|q
i
ti › H
(16.54)