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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