Covariant Derivatives and the Hamilton-Jacobi Equation
Sabrina Gonzalez Pasterski
(Dated: March 2, 2014)
I define a covariant derivative to simplify how higher order derivatives act on a classical generating
function.
When studying the connection between classical and
quantum mechanics, it would be nice to have a differential operator which, when acting repeatedly on some
S(q,P,t)
pulls down powers of the derivatives of
function ei ~
the function within the exponent.
Consider the results of “Wavefunctions and the
Hamilton-Jacobi Equation.” There, I performed a canonical change of variables from (qi , pi ) to constants (Qi , Pi ):
dF
pi q˙i − H(q, p, t) = Pi Q˙ i − K(Q, P, t) +
dt
n S(x,P,t)
S(x,P,t)
00
since Γxxx = SS 0 , so ~i S 0 ei ~
= pˆn ei ~
=
S(x,P,t)
~ n
i
~
∇
e
holds
by
induction.
n
i
If I define a partition function expectation value:
R
hO(x, p)i ≡
S(x,P,t)
P(P )dP dqO(x, p)ei ~
R
S(x,P,t)
P(P )dP dqei ~
(1)
then this is equivalent to:
(2)
hO(x, p)i =
where F = S(q, P, t) − Pi Qi , and found:
∂S
H=−
∂t
∂S
pi =
∂qi
R
∂S
Qi =
.
∂Pi
S(x,P,t)
~
ˆ pˆ) : ei
P(P )dP dx : O(x,
R
S(x,P,t)
P(P )dP dxei ~
(7)
(8)
S
when K = 0. At first order, the function ei ~ had the
property that ordinary multiplication by the value of H
or pj was equivalent to acting with a differential operator:
S
S
S
H · ei ~ = − ∂S
ei ~ = i~∂t ei ~
∂t
(3)
S
pj · ei ~ =
∂S
∂qj
S
S
ei ~ = ~i ∂qj ei ~ .
Moreover, this connection between multiplication and a
differential operator held at first order for arbitrary superpositions:
Z
Φ(q, t) =
P(Pi )ei
S(q,P,t)
~
dPi .
S(x,P,t)
~
h dO
i = h{O, H}i
dt
\
= h: {O,
H} :i
ˆ :, : H
ˆ :] :i
= −i h:[: O
(4)
In this paper, I will consider the one dimensional case
q = x and define a covariant derivative such that acting
S(x,P,t)
on ei ~
a total of n times with the operator ~i ∇x
is exactly equivalent to multiplying ei
n
( ∂S
∂x ) .
where the normal ordered operator is defined such that
all of the momentum operators appear on the right. The
direct correspondence between xn pm = xn (S 0 )m in O
ˆ pˆ) : thus follows from
and xn pˆm = xn ( ~i )m ∇m in : O(q,
the composition property of the covariant derivative.
ˆ pˆ) :i, one
Summarizing Equation 8 as hO(q, p)i = : O(q,
finds that for an operator which does not explicitly depend on time:
by pn =
(9)
~
The last equality comes from considering a generic term
in the series expansion of O(x, p)
S(x,P,t)
The key is to treat ei ~
as a scalar, with a non2
trivial one-dimensional spatial metric gxx = ( ∂S
∂q ) . Then
h{xn pm , xr ps }i = (ns − mr)hxn+r−1 pm+s−1 i
= (ns − mr)hxn+r−1 pˆm+s−1 i
(10)
00
there is a non-zero connection Γxxx = 12 g xx ∂x gxx = SS 0 ,
where primes denote partial derivatives with respect to
x.
n
S(x,P,t)
S(x,P,t)
If I treat pˆn ei ~
≡ ~i ∇n ei ~
as a covariant
rank-n tensor, I find that:
S
S
∇n ei ~ ≡ ∇x ∇x ...∇x ei ~
S
S
= ∂x (∇n−1 ei ~ ) − (n − 1)Γxxx ∇n−1 ei ~
S
S
−i
h:[xn pˆm , xr pˆs ] :i
~
= h: xn [ˆ
pm , xr ]ˆ
ps + xr [xn , pˆs ]ˆ
pm :i
n+r−1 m+s−1
= (ns − mr)hx
pˆ
i
(11)
(5)
S
It is quick to check for n = 1 that ∇1 ei ~ ≡ ∇x ei ~ =
S
S
S
∂x ei ~ . If it is true that ∇n−1 ei ~ = ( ~i S 0 )(n−1) ei ~ , then:
S
versus
S
∇n ei ~ = ∂x (( ~i S 0 )(n−1) ei ~ ) − (n − 1)Γxxx · ( ~i S 0 )(n−1) ei ~
S
S
= (n − 1)( ~i S 0 )(n−2) ~i S 00 ei ~ + ( ~i S 0 )n ei ~
S
− (n − 1)Γxxx · ( ~i S 0 )(n−1) ei ~
S
= ( ~i S 0 )n ei ~
(6)
where some care must be taken when specifying what it
means to normal order the commutator (e.x. I would
want to have : [x, pˆ] := i~ and not : [x, pˆ] :=: xˆ
p : − :
pˆx := 0).
Equation 9 is similar to Ehrenfest’s Theorem. There
is a natural association between the Poisson Bracket of
classical mechanics and the normal ordered commutator
of normal ordered operators.
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