3. Feynman and Jacobi path integrals

The path integral approach to string-dynamics is not only a useful check of the quantization procedure, but gives a new insight into the quantum theory itself. In particular, it gives a better insight into the meaning of the equivalence between the Nambu-Goto action and the Schild action. Furthermore, it provides a physically transparent relationship between the Feynman ``sum over histories'' integral and the Jacobi path integral. However, to see how this comes about, one must keep in mind that the momenta $p _{\mu \nu}$ and $\pi _{ab}$ cannot vary freely, because $\pi _{ab}$ is merely a shorthand notation for the function $\epsilon _{ab} H (p)$, and thus must satisfy the constraint equation

$$\pi _{ab} - \epsilon _{ab} H (p) = 0 \ .$$ (13)

This constraint may be incorporated in the action (3) by means of a Lagrange multiplier $N ^{ab} (\sigma)$

$$S [ x (\sigma) , p (\sigma) , \xi (\sigma) , N (\sigma) ; A] = \frac{1}{2} \int _{X ( \sigma )} p _{\mu \nu} d x ^{\mu} \wedge d x ^{\nu} +$$

$$\qquad \qquad + \frac{1}{2} \int _{\xi (\sigma)} \pi _{ab} d \xi ... ...{2} \sigma N ^{ab} (\sigma) \left[ \pi _{ab} - \epsilon _{ab} H (p) \right] \ ,$$ (14)

and its physical meaning can be read out of the classical equation of motion obtained by varying $\pi _{ab}$:

$$N ^{ab} (\sigma) = \epsilon ^{mn} \partial _{m} \xi ^{a} \partial _{n} \xi ^{b} \ .$$ (15)

Thus, N ^ab is the transformed $\epsilon$-tensor in the new coordinate system. Then, apart from an over all normalization constant ⁴, the amplitude for the initial string C ₀ to ``evolve'' into the final string C in a lapse of ``time A'', can be represented by the path integral

 

$$K [ x (s) , x _{0} (s) ; A ] = \int _{ x _{0} (s)} ^{x (s)} \int ... ...(s)} ^{\xi (s)} [D \mu (\sigma)] e ^{i S [ x , p , \xi , \pi , N ; A ] / \hbar}$$

$$[D \mu (\sigma)] \equiv [D x ^{\mu} (\sigma)] [D \xi ^{a} (\sigma)] [D p _{\mu \nu} (\sigma)] [D \pi _{ab} (\sigma)] [D N _{cd} (\sigma)] \ .$$ (17)

At the classical level, H, or $\pi _{ab}$, is independent of the coordinates $\sigma ^{m}$ because of the balance equation (4). The same result is obtained at the quantum level by integrating out the $\xi ^{a} ( \sigma )$ fields:

 

$$\int _{\xi _{0} (s)} ^{\xi (s)} [D \xi ^{a} (\sigma)] \exp \left\... ...\hbar} \int _{\xi (\sigma)} d \xi ^{a} \wedge d\xi ^{b} \, \pi _{ab} \right\} =$$

$$\qquad \qquad = \delta \left[ \epsilon ^{mn} \partial _{m} \pi _{... ...ar} \int _{\xi (\sigma)} d \left( \pi _{ab} \xi ^{a} d \xi ^{b} \right) \quad .$$ (18)

The functional Dirac-delta requires $\pi _{ab}$ to satisfy the classical equation of motion, i.e. $\pi _{ab} = \epsilon _{ab} \times \mathrm{const.} \equiv \epsilon _{ab} E$. Therefore,

$$\int \! [D \pi _{ab}] \delta \left[ \epsilon ^{mn} \partial _{m} ... ... _{\Sigma} \! \! \! d ^{2} \sigma N ^{ab} (\sigma) \pi _{ab} \right] \right\} =$$

$$\qquad \qquad \qquad = \int _{0} ^{\infty} d E e ^{i E A / \hbar}... ...{2 \hbar} \int _{\Sigma} d ^{2} \sigma N ^{ab} (\sigma) \epsilon _{ab} \right\}$$ (19)

where we have assumed that the Hamiltonian is bounded from below and is normalized in such a way that $E \ge 0$. Then, the Feynman path integral can be written as follows

 

$$K[ x (s) , x _{0} (s) ; A ] = \int _{0} ^{\infty} d E e ^{i E A /... ... (s)} [D x ^{\mu} (\sigma)] [Dp _{\mu \nu} (\sigma)] [DN _{cd} (\sigma)] \times$$

$$\times \exp \left\{ \frac{i}{2 \hbar} \int _{X (\sigma)} p _{\mu ... ...} \int _{\Sigma} d ^2 \sigma N ^{ab} (\sigma) \left[ E - H (p) \right] \right\}$$

$$\qquad \qquad \qquad \quad \, \, \, \equiv 2 i \hbar m ^{2} \int _{0} ^{\infty} d E e ^{ i E A / \hbar} G[ C , C _{0} ; E] \ ,$$ (20)

where we have introduced the Jacobi path integral G[ C , C ₀ ; E] as the amplitude for a string to propagate from C ₀ to C, at fixed energy E.

The most important property of G[ C , C ₀ ; E] is reparametrization invariance. If we integrate out the area momentum $p _{\mu \nu}$, we obtain

 

$$= \int _{x _{0} (s)} ^{x (s)} [D x ^{\mu} (\sigma)][D N (\sigma)] \times$$ G[ C , C ₀ ; E ]

$$\qquad \qquad \qquad \times \exp \left\{ - \frac{i}{\hbar} \int _... ...rac{m ^{2}}{4 N} {\dot{x}}^{\mu \nu} {\dot{x}}_{\mu \nu} + N E \right] \right\}$$ (21)

where $N (\sigma) \equiv \epsilon _{ab} N ^{ab} (\sigma) / 2$.
Equation (21) is manifestly invariant under reparametrization

$${\dot{x}}^{\mu \nu} (\sigma) \rightarrow \mathrm{det} \le... ...artial u ^{a}}{\partial \sigma ^{m}} \right) N (\sigma) \ .$$ (22)

Finally, if we estimate the path integral (21) around the saddle point $\widehat N (\sigma) = \left( - m ^{2} {\dot{x}}^{\mu \nu} {\dot{x}}_{\mu \nu} / 4 E \right) ^{1/2}$, we find

$$G[ C , C _{0} ; E ] = \int _{ x _{0} (s)} ^{x (s)} [ D x ^{... ...sqrt{- {\dot{x}}^{\mu \nu} {\dot{x}}_{\mu \nu}} \right\} \ ,$$ (23)

which is the usual path integral weighed by the Nambu-Goto action, once we fix E = m ² / 2.
The result (23) suggests the following concluding remarks for this section:

1.

equation (23) could be assumed at the outset and taken as a starting point for string quantization by means of functional techniques. The advantage of our derivation is that it clarifies the physical meaning of such a reparametrization invariant path integral: it represents the string propagation amplitude at fixed ``area-energy'' $E = 1/4 \pi \alpha '$.

2.

We can invert the Fourier transform (20) and define the reparametrization invariant path integral in terms of the Feynman propagation amplitude at fixed ``area-lapse'' A

$$G[ C , C _{0} ; m ^{2} ] \equiv \frac{1}{2 i \hbar m ^{2}} ... ... e ^{- i m ^{2} A / 2 \hbar} K[ x (s) , x _{0} (s) ; A] \ .$$ (24)

Then, reparametrization offers an alternative definition of the sum over histories: first, sum over all world-sheets of fixed area; then, integrate over all possible values of the world-sheet area.

3.

Equation (20) represents the quantum counterpart of the classical equivalence [5] between the Nambu-Goto action and the Schild action.