World-Line Path Integral for the Propagator Expressed as an Ordinary Integral: Concept and Applications

Abstract

The (Feynman) propagator $G(x_2,x_1)$ encodes the entire dynamics of a massive, free scalar field propagating in an arbitrary curved spacetime. The usual procedures for computing the propagator—either as a time ordered correlator or from a partition function defined through a path integral—requires introduction of a field $\phi (x)$ and its action functional $A[\phi (x)]$. An alternative, more geometrical, procedure is to define a propagator in terms of the world-line path integral which only uses curves, $x^i(s)$, defined on the manifold. I show how the world-line path integral can be reinterpreted as an ordinary integral by introducing the concept of effective number of quantum paths of a given length. Several manipulations of the world-line path integral becomes algebraically tractable in this approach. In particular I derive an explicit expression for the propagator $G_\mathrm{QG}(x_2,x_1)$, which incorporates the quantum structure of spacetime through a zero-point-length, in terms of the standard propagator $G_\mathrm{std}(x_2,x_1)$, in an arbitrary curved spacetime. This approach also helps to clarify the interplay between the path integral amplitude and the path integral measure in determining the form of the propagator. This is illustrated with several explicit examples.

Appendix: Calculational Details

1.1 Path Measure in Real Space

Given the path measure in the momentum space, $N_\mathrm{free}(\ell , p) = \cos p\ell$, we can find the measure $N_\mathrm{free}(\ell , x)$ in real space by evaluating the D-dimensional Fourier transform of $\cos p\ell$. To do this, we start with the standard result for the Fourier transform of spherically symmetric function. If

$$\begin{aligned} F(k) = \int d^D\varvec{x}\ f(|\varvec{x}|) \, e^{-i\varvec{k\cdot x}} \end{aligned}$$

(42)

then we can write

$$\begin{aligned} k^{\frac{D-2}{2}} F(k) = (2\pi )^{D/2} \int _0^\infty J_{\frac{D-2}{2}} (kr) \, r^{\frac{D-2}{2}}\ f(r)\, r \, dr \end{aligned}$$

(43)

This allows us to write the relevant Fourier transform as:

$$\begin{aligned} N_{free}(\ell ,x) = \int _0^\infty \frac{d^Dp}{(2\pi )^D}\ e^{ip.x} \cos p\ell = \frac{1}{(2\pi )^{D/2}} \frac{1}{x^\alpha } \int _0^\infty dp \ p^{\alpha + 1} J_\alpha (p x) \cos \ell p \end{aligned}$$

(44)

where $\alpha = (D/2) - 1$. We next evaluate this integral using the standard cosine transform (see, for e.g., page 45, 1.12 (12) of [27])

$$\begin{aligned} \int _0^\infty dx\ (\cos xy)\, x^{\nu + 1} J_\nu (ax) = \frac{2^{\nu + 1} \sqrt{\pi }\, a^\nu y \Theta [y -a]}{\Gamma \left( -\frac{1}{2} - \nu \right) (y^2 - a^2)^{\nu + \frac{3}{2}}} \end{aligned}$$

(45)

This gives the result

$$\begin{aligned} N_{free}(\ell ,x) = \frac{1}{\pi ^{\frac{D-1}{2}}} \ \frac{\Theta [\ell ^2-x^2]}{\Gamma \left( -\frac{D-1}{2}\right) }\ \frac{\ell }{[\ell ^2 - x^2]^{\frac{D+1}{2}}} \end{aligned}$$

(46)

which reduces to the expression quoted in the text when $D=4$. (Strictly speaking the integral in Eq. (45) is defined only for $-1<\mathrm{Re}\nu <-{1/2}$ but can be analytically continues for other $\nu$, as often done in dimensional regularization.) Note that $N_\mathrm{free}(\ell , x)$ vanishes for $\ell < x$; paths, with lengths less than the geometrical length between the two points, do not contribute to the path integral which is rather nice feature.

One can directly verify that this expression leads to the correct massive propagator in real space which, of course, is obvious from the fact that $N_\mathrm{free}(\ell , x)$ is the Fourier transform of $N_\mathrm{free} (\ell ,p)$ and we know that the latter gives the correct propagator in the momentum space. To verify this directly we need the integral

$$\begin{aligned} \int _1^\infty dt\ \frac{t \ e^{-zt}}{(t^2-1)^{\frac{1}{2} - \nu }} = \frac{\Gamma \left( \nu + \frac{1}{2}\right) }{\sqrt{\pi }}\, \left( \frac{2}{z}\right) ^\nu \, K_{\nu +1}(z) \end{aligned}$$

(47)

which can be obtained from a standard result (see, 8.432 (3) of [28]) by differentiating with respect to z and using the recursion relation for $K_\nu (z)$ (see page 929 (13) of [28]). Using this integral and the expression for $N_{free}(\ell ,x)$ in Eq. (46). we find that (with $\mu \equiv (1/2) (D+1)$)

$$\begin{aligned} \int _0^\infty d\ell \ N_\mathrm{free}(\ell ,x)\ e^{-m\ell }&= \int _x^\infty d\ell \frac{1}{\pi ^{\mu -1}} \, \frac{1}{\Gamma (1-\mu )} \, \frac{\ell \ e^{-m\ell }}{(\ell ^2 - x^2)^\mu }\nonumber \\&= \frac{x^{2(1-\mu )}}{\pi ^{\mu -1}\sqrt{\pi }} \, \left( \frac{2}{mx}\right) ^{\frac{1}{2} - \mu } K_{\mu - \frac{3}{2}} (m x) = \mathcal {G}_D(mx) \end{aligned}$$

(48)

which is indeed m times the standard massive propagator $\mathcal {G}_D(mx) =mG_D(mx)$ in D-dimensions. In the case of $D=4$, this reduces to

$$\begin{aligned} \mathcal {G}(x) = m \, G(x) = \frac{m}{4\pi ^2} \left( \frac{m}{x}\right) \, K_1(mx) \end{aligned}$$

(49)

which is the familiar expression.

1.2 Relation Between $\mathcal {G}_{QG}$ and $\mathcal {G}_{std}$

I will now sketch the proof of Eq. (18) and Eq. (16) which involves slightly nontrivial manipulations of integrals over Bessel functions. I begin with two easily provable identities:

$$\begin{aligned} \frac{e^{-m\sqrt{\ell ^2 + L^2}}}{\sqrt{\ell ^2 + L^2}} = \int _m^\infty dm_0\ e^{-m_0\ell } \, J_0\left[ L \sqrt{m_0^2 - m^2}\right] \end{aligned}$$

(50)

and

$$\begin{aligned} \frac{1}{2s} e^{-m^2s - \frac{L^2}{4s}} = \int _m^\infty dm_0\ m_0 e^{-m_0^2s} J_0 \left[ L \sqrt{m_0^2-m^2}\right] \end{aligned}$$

(51)

The first identity in Eq. (50) can be proved by changing the integration variable on the right hand side to $x\equiv m_0/m$ and using a standard integral (see, 6.616 (2) of [28])

$$\begin{aligned} \int _1^\infty dx\ e^{-\alpha x} J_0\left[ \beta \sqrt{x^2-1}\right] = \frac{e^{-\sqrt{\alpha ^2 + \beta ^2}}}{\sqrt{\alpha ^2 + \beta ^2}} \end{aligned}$$

(52)

The second identity in Eq. (51) can again be proved by changing the integration variable on the right hand side to $x=m_0/m$ and using a result derivable from 6.614 (1) of [28]:

$$\begin{aligned} \int _0^\infty 2k dk \ J_0(kL) e^{-sk^2} = \frac{1}{s} \exp \left( -\frac{L^2}{4s}\right) \end{aligned}$$

(53)

Differentiating both sides of Eq. (50) with respect to m, we obtain

$$\begin{aligned} e^{-m\sqrt{\ell ^2 + L^2}}&= -\frac{\partial }{\partial m} \int _m^\infty dm_0\ e^{-m_0 \ell } J_0\left[ L \sqrt{m_0^2-m^2}\right] \nonumber \\&= \int _0^\infty dm_0 \ e^{-m_0\ell }\, (-1) \frac{\partial }{\partial m} \left\{ \Theta (m_0-m) J_0\left[ L \sqrt{m_0^2 - m^2}\right] \right\} \end{aligned}$$

(54)

This gives

$$\begin{aligned} e^{-m\sqrt{\ell ^2 + L^2}} = \int _0^\infty dm_0\ e^{-m_0\ell } \mathcal {P}[m_0; m, L] \end{aligned}$$

(55)

with $\mathcal {P}$ defined by Eq. (19). Similarly, differentiating both sides of Eq. (51) with respect to m and manipulating as before we get

$$\begin{aligned} m e^{-m^2 s - \frac{L^2}{4s}} = \int _0^\infty dm_0 \ m_0 e^{-m_0^2 s } \mathcal {P} [m_0; m,L] \end{aligned}$$

(56)

Notice that the right hand sides of Eqs. (56) and  (55) have very similar structures with $e^{-m_0\ell }$ replaced by $m_0e^{-m_0^2 s}$.

The results in Eqs. (18) and  (17), which we need to prove, can now be obtained in a straightforward manner as follows: Multiply both sides of Eq. (55) by $N(\ell )$ and integrate over $\ell$ to get:

$$\begin{aligned} \mathcal {G}_\mathrm{QG}&= \int _0^\infty d\ell \ N(\ell ) \, e^{-m\sqrt{\ell ^2 + L^2}} = \int _0^\infty dm_0\ \mathcal {P} [ m_0; m, L] \int _0^\infty d\ell \ N(\ell ) \, e^{-m_0 \ell }\nonumber \\&= \int _0^\infty dm_0\ \mathcal {P} [ m_0; m, L]\mathcal {G}_\mathrm{std} (m_0) \end{aligned}$$

(57)

where $\mathcal {P} [ m_0; m, L]$ is defined by Eq. (19), reproduced here for ready reference:

$$\begin{aligned} \mathcal {P}[m_0; m,L] = - \frac{\partial }{\partial m} \left\{ \Theta ( m_0-m) J_0\left[ L \sqrt{m_0^2 - m^2}\right] \right\} \end{aligned}$$

(58)

This gives Eq. (18). To obtain Eq. (17), we write $\mathcal {G}_\mathrm{std}$ in terms of $K_\mathrm{std}(s)$. This leads to

$$\begin{aligned} \mathcal {G}_\mathrm{QG}&= \int _0^\infty dm_0\ \mathcal {P} [ m_0; m, L]\int _0^\infty ds\ e^{-m_0^2s} K_{std}(s) m_0 \nonumber \\&= \int _0^\infty ds\ K_{std}(s)\int _0^\infty dm_0\ m_0 \, e^{-m_0^2s}\mathcal {P} [ m_0; m, L]\nonumber \\&= \int _0^\infty ds\ K_{std}(s) m e^{-m^2 s - \frac{L^2}{4s}} \end{aligned}$$

(59)

In arriving at the last equality we have used the result in Eq. (56). This proves Eq. (17).

To obtain the relation between $G_\mathrm{QG} = \mathcal {G}_\mathrm{QG} / m$ and $G_\mathrm{std} = \mathcal {G}_\mathrm{std} / m$ we can proceed as follows. We pull the derivative with respect to m (arising from the expression in Eq. (58)) out of the integral sign in Eq. (57) and change the variable of integration to k with $k^2 \equiv m_0^2 - m^2$. This leads to the relation:

$$\begin{aligned} \mathcal {G}_\mathrm{QG} (m) = - \frac{\partial }{\partial m} \int _0^\infty \frac{kdk}{\sqrt{k^2+m^2}}\ J_0(Lk)\ \mathcal {G}_\mathrm{std}(m^2+k^2) \end{aligned}$$

(60)

where $\mathcal {G}_\mathrm{std}(m^2+k^2)$ is the standard propagator with $m^2$ replaced by $m^2+k^2$. Using the fact that $\mathcal {G}(M) = MG(M)$ for any mass parameter M, this is equivalent to

$$\begin{aligned} G_\mathrm{QG} (m)&= - \frac{1}{m}\frac{\partial }{\partial m} \int _0^\infty \frac{kdk}{\sqrt{k^2+m^2}}\ J_0(Lk)\ \sqrt{k^2+m^2} \, G_\mathrm{std}(m^2+k^2)\nonumber \\&= - \frac{\partial }{\partial m^2} \int _0^\infty 2k\, dk\ J_0(Lk)\ G_\mathrm{std} ( k^2 +m^2) \end{aligned}$$

(61)

This is the relation quoted in the main text; see Eq. (22). Further, for any function f(k) which depends only on the magnitude of the 2-dimensional vector $\varvec{k}$, we have the identity

$$\begin{aligned} \int \frac{d^2k}{(2\pi )^2} \, e^{i\varvec{k\cdot L}}\, f(k) = \frac{1}{2\pi }\int _0^\infty k\,dk\ J_0(kL)\, f(k) \end{aligned}$$

(62)

Using this Eq. (61) can be expressed in the form

$$\begin{aligned} G_\mathrm{QG}(m) = - 4\pi \frac{\partial }{\partial m^2} \int \frac{d^2k}{(2\pi )^2}\, e^{i\varvec{k\cdot L}}\, G_\mathrm{std} ( k^2 +m^2) \end{aligned}$$

(63)

where $\varvec{L}$ is a 2-dimensional vector with magnitude equal to the zero-point-length. From this it is possible to obtain a higher dimensional interpretation of $G_\mathrm{QG}$. However, the result is probably more transparent when obtained from first principles and I will provide such a derivation:

I will now work in D dimensional space(time) and express $G_\mathrm{QG}^D$ in D-dimensions in terms of the standard propagator $G_\mathrm{std}^N$ in $N=D+2$ dimensions. To do this, let us consider a fictitious $N=D+2$, Euclidean curved space(time) with the metric

$$\begin{aligned} dS_N^2 = \left( g_{ab} dx^a dx^b\right) _D + \delta _{AB}\ dX^A dX^B \qquad \qquad\quad (A,B = 1,2) \end{aligned}$$

(64)

where we have added two “flat” directions, $X^A$ with $A=1,2$. (The metric $g_{ab}$ in D dimensions, of course depends only on the D coordinates $x^a$.) The N-dimensional Schwinger kernel for $m=0$ (ZMSK) now factorizes and we can write

$$\begin{aligned} K^N_{m=0} \equiv {\langle x,\varvec{L}|e^{s\Box _N}|y,\varvec{0}\rangle } =\left( \frac{1}{4\pi s}\right) e^{-L^2/4s} \, K^D_{m=0}(x,y; s) \, ;\qquad L^2 \equiv L_AL^A \end{aligned}$$

(65)

Therefore, the corresponding N-dimensional, massive, Schwinger kernel becomes

$$\begin{aligned} K^N(x,\varvec{L}; y,\varvec{0}; s) = \left( \frac{1}{4\pi s}\right) e^{-(L^2/4s) - m^2s} \, K^D_{m=0}(x,y; s) \end{aligned}$$

(66)

The corresponding N-dimensional massive propagator is obtained by integrating this kernel over s in the range 0 to $\infty$. This is almost the same as $G_\mathrm{QG}$ but for the extra factor $(1/4\pi s)$. This factor can be taken care of by differentiating the propagator with respect to $m^2$. We then find that

$$\begin{aligned} -4\pi \frac{\partial }{\partial m^2} \, G_\mathrm{std}^N(x,\varvec{L}; y,\varvec{0}) = \int _0^\infty ds\ e^{-(L^2/4s) - m^2s} \, K^D_{m=0}(x,y; s) = G_\mathrm{QG}^D (x,y) \end{aligned}$$

(67)

So, we have related the quantum corrected propagator $G_\mathrm{QG}^D(x,y)$ in D dimensions to the standard Klein-Gordan propagator in the fictitious $N=D+2$ space with the metric in Eq. (64), through the relation:

$$\begin{aligned} G_\mathrm{QG}^D(x,y) = - 4\pi \frac{\partial }{\partial m^2} G^N_\mathrm{std}(x,\varvec{L}; y, \varvec{0}) \bigg |_{\mathbf {L}^2=L^2} \end{aligned}$$

(68)

The $N=D+2$ dimensional propagator, $G^N_\mathrm{std}(x,\varvec{L}; y, \varvec{0})$, of course has a standard QFT interpretation in the curved spacetime. The zero-point-length in D dimensions arises as the magnitude of the (fictitious) propagation distance in the extra dimensions. This proves Eq. (24) in the main text.

The derivative of standard Klein-Gordan propagator with respect to $m^2$—which appears in the right hand side of Eq. (68) —can be related to the ‘transitivity integral’ for the propagator $G^N_\mathrm{std}$. To see this, consider the following integral (with the notation $d^N\bar{z} \equiv d^Dz\, d^2Z$):

$$\begin{aligned}&\int d^N\bar{z} \, G^N_\mathrm{std}(x,\varvec{L}; z,\varvec{Z}) \, G^N_\mathrm{std}(z,\varvec{Z}; y,\varvec{0}) \nonumber \\&\quad = \int d^N\bar{z} \ {\langle x,\varvec{L}|(-\Box ^N + m^2)^{-1}|z,Z\rangle } {\langle z,Z|(-\Box ^N + m^2)^{-1}|y,\varvec{0}\rangle }\nonumber \\&\quad = {\langle x,\varvec{L}|(-\Box ^N + m^2)^{-2}|y,\varvec{0}\rangle } \end{aligned}$$

(69)

We can, however, write:

$$\begin{aligned} {\langle x,\varvec{L}|(-\Box ^N + m^2)^{-2}|y,\varvec{0}\rangle } = - \frac{\partial }{\partial m^2} \, G^N_\mathrm{std}(x,\varvec{L}; y,\varvec{0}) \end{aligned}$$

(70)

Combining this result with Eq. (68) we find that

$$\begin{aligned} G^D_{QG} = (4\pi ) {\langle x,\varvec{L}|(-\Box ^N + m^2)^{-2}|y,\varvec{0}\rangle } \bigg |_{\mathbf {L}^2=L^2} \end{aligned}$$

(71)

This result, in turn, can expressed in the form of either of the following differential equations for $G^D_{QG}$, viz.:

$$\begin{aligned} (-\Box ^N + m^2)^{2} G^D_{QG} =4\pi \delta (x,y)\, \delta (\varvec{L}) \end{aligned}$$

(72)

and

$$\begin{aligned} (-\Box ^N + m^2) G^D_{QG} = 4\pi G^N_\mathrm{std}(x, \varvec{L}; y,\varvec{0}) \end{aligned}$$

(73)

These are valid any curved space(time). We can now write $G^N_\mathrm{std}(x, \varvec{L}; y,\varvec{0})$ as the standard vacuum expectation value of time ordered products of the KG field operators in $N=D+2$ space(time); then the QG corrected propagator in D dimensional space is given by a solution to Eq. (73). This proves Eqs. (25) and  (26).

We can, of course, verify that, in flat space(time) either of these equations lead to the correct $G_\mathrm{QG}$. For example, if we Fourier transform either Eqs. (72) or  (73), we will find that $G^D_\mathrm{QG}$ can be expressed as the integral

$$\begin{aligned} G^D_{QG} (x, \varvec{L}; 0,\varvec{0}) = 4\pi \int \frac{d^Dk \, d^2K}{(2\pi )^N} \ \frac{e^{ikx}\, e^{i\varvec{K\cdot L}}}{(k^2 + m^2 + K^2)^2} \end{aligned}$$

(74)

The square appearing in the denominator can be taken care of by the usual trick of differentiating the expression with respect to $m^2$. Performing the 2-dimensional integral over the measure $d^2K = K dK d\theta$ we get the result in terms of the Bessel function $J_0(KL)$:

$$\begin{aligned} G^D_{QG} (x, \varvec{L}; 0,\varvec{0})&= -\frac{\partial }{\partial m^2} \int \frac{d^Dk}{(2\pi )^D} \int _0^\infty K dK\int _0^{2\pi }\frac{d\theta }{\pi } \, \frac{e^{ikx}\, e^{i{K L\cos \theta }}}{k^2 + m^2 + K^2} \nonumber \\&= -\frac{\partial }{\partial m^2} \int \frac{d^Dk}{(2\pi )^D}\, e^{ikx}\int _0^\infty 2K dK\ \frac{J_0(KL)}{k^2 + m^2 + K^2} \end{aligned}$$

(75)

Therefore the D-dimensional Fourier transform $G^D_\mathrm{QG} (k,\varvec{L})$ of $G^D_\mathrm{QG}(x, \varvec{L}; 0,\varvec{0})$ is given by

$$\begin{aligned} G^D_\mathrm{QG} (k,\varvec{L}) = - \frac{\partial }{\partial m^2} \int _0^\infty 2K dK \ \frac{J_0(KL)}{k^2 + m^2+K^2} \end{aligned}$$

(76)

To perform the integral over K we write the denominator in the exponential form, leading to:

$$\begin{aligned} -\frac{\partial }{\partial m^2} \int _0^\infty 2K dK&\ J_0(KL) \int _0^\infty ds\ e^{-s(k^2+m^2)} \, e^{-sK^2}\nonumber \\&= \int _0^\infty ds\ s e^{-s(k^2+m^2)} \times \ \int _0^\infty 2K \, dK\ J_0(KL) e^{-sK^2} \end{aligned}$$

(77)

Finally, we use the identity

$$\begin{aligned} \int _0^\infty 2K dK \ J_0(KL) e^{-sK^2} = \frac{1}{s} \exp \left( -\frac{L^2}{4s}\right) \end{aligned}$$

(78)

to recover the standard result

$$\begin{aligned} G^D_\mathrm{QG} (k,{L}) = \int _0^\infty ds\ e^{-s(k^2+m^2) - (L^2/4s)} \end{aligned}$$

(79)