Principle of equivalence at Planck scales, QG in locally inertial frames and the zero-point-length of spacetime

Abstract

Principle of equivalence makes effects of classical gravity vanish in local inertial frames. What role does the principle of equivalence play as regards quantum gravitational effects in the local inertial frames? I address this question here from a specific perspective. At mesoscopic scales close to, but somewhat larger than, Planck length one could describe quantum spacetime and matter in terms of an effective geometry. The key feature of such an effective quantum geometry is the existence of a zero-point-length. When we proceed from quantum geometry to quantum matter, the zero-point-length will introduce corrections in the propagator for matter fields in a specific manner. On the other hand, one cannot ignore the self gravity of matter fields at the mesoscopic scales and this will also modify the form of the propagator. Consistency demands that, these two modifications—coming from two different directions—are the same. I show that this non-trivial demand is actually satisfied. Surprisingly, the principle of equivalence, operating at sub-Planck scales, ensures this consistency in a subtle manner.

Appendices

Appendix: Calculational details

This “Appendix” gives the details of some of the calculations as well as some alternative derivations and extensions.

Relation between $G_{QG}$ and $G_{std}$

The inclusion of zero-point-length modifies standard (rescaled) propagator

$$\begin{aligned} {\mathcal {G}}_{std}(x,y;m)\equiv mG_{std}=\int _0^\infty m\ ds\ e^{-m^2s}K_{std}(s; x,y) \end{aligned}$$

(22)

to the form ${\mathcal {G}}_{QG}$ which incorporates the quantum corrections:

$$\begin{aligned} {\mathcal {G}}_{QG}(x,y;m)=\int _0^\infty m\ ds\ e^{-m^2s-L^2/4s}K_{std}(s; x,y) \end{aligned}$$

(23)

There are two ways of understanding this result. The simple, intuitive way is to recall that the leading order behaviour of the heat kernel is $K_{\mathrm{std}} \sim s^{-2} \exp [-\sigma ^2(x,y)/4s]$ where $\sigma ^2$ is the geodesic distance between the two events; so the modification in Eq. (23) amounts to the replacement $\sigma ^2 \rightarrow \sigma ^2 +L^2$ to the leading order. This gives the leading QG corrections to the propagator at mesoscopic scales. The corrections due to background curvature, captured in the Schwinger-DeWitt coefficients are irrelevant at the mesoscopic scales $\lambda$ with $L_P\lesssim \lambda \ll L_{curv}$; this is what I called the flat spacetime quantum gravity regime.

More rigorously, one can arrive at Eq. (23) from the principle of path integral duality. This principle postulates [1, 2] that the effect of zero-point-length is to modify the path integral to the form:

$$\begin{aligned} {\mathcal {G}}_{QG}(x,y;m) = \sum _\sigma \exp \left[ - m \left( \sigma + \frac{L^2}{\sigma }\right) \right] =\int _0^\infty m\ ds\ e^{-m^2s-L^2/4s}K_{std}(s; x,y) \nonumber \\ \end{aligned}$$

(24)

The path integral sum can be computed by lattice regularization techniques [1, 2] and will lead to the second equality.

I will briefly outline how straight forward algebra allows one to relate $G_{QG}$ and $G_{std}$. We start with a standard integral involving Bessel function:

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

(25)

and obtain from this the result

$$\begin{aligned} \int _a^\infty dt \ e^{-pt} J_0\left[ 2 \sqrt{b(t-a)}\right] = \frac{1}{p} e^{-ap - (b/p)} \end{aligned}$$

(26)

which can be verified by setting $b(t-a) = x^2$ and using Eq. (25). Differentiating both sides of Eq. (26) with respect to a we get

$$\begin{aligned} e^{-ap - (b/p)} = - \frac{\partial }{\partial a} \int _a^\infty dt\ e^{-pt} J_0\left[ 2 \sqrt{b(t-a)}\right] \end{aligned}$$

(27)

The limits of integration in the right hand side can be extended from 0 to $\infty$ by introducing a factor $\theta [t-a]$ in the integrand. Moving the differential operator $\partial /\partial a$ inside the integral, one will then obtain one term containing $\theta J_1$ and another term of the form $J_0 \delta$ giving rise to $e^{-ap}$. It turns out, however, more convenient not to do this and instead use the expression in Eq. (27) as it is in the computations. The differentiation is best carried out towards the end, when required. I will now set $a=m^2$ and $b=L^2/4$ in Eq. (27) to obtain:

$$\begin{aligned} e^{-m^2s - (L^2/4s)}&= - \frac{\partial }{\partial m^2} \int _{m^2}^\infty dt\ e^{-st} J_0\left[ L \sqrt{t-m^2}\right] \nonumber \\&=- \frac{\partial }{\partial m^2}\int _{m^2}^\infty d m_0^2\ e^{-m_0^2s} J_0\left[ L \sqrt{m_0^2-m^2}\right] \end{aligned}$$

(28)

where, in the second step, I have put $t=m_0^2$. I insert this expansion in the definition of quantum gravitational propagator, given by

$$\begin{aligned} G_{\mathrm{QG}} (m^2) = \int _0^\infty ds\ e^{-m^2s - (L^2/4s)} K_0(s) \end{aligned}$$

(29)

where $K_0(s) = {\langle x|e^{s\Box }|y\rangle }$ is the zero-mass heat kernel in an arbitrary curved space(time)⁸ with $\Box \equiv \Box _g$ being the Laplacian corresponding to the curved space metric $g_{ab}$. Using Eq. (28) in Eq. (29) I obtain:

$$\begin{aligned} G_{\mathrm{QG}}(m^2)&= - \int _0^\infty ds \ K_0(s) \frac{\partial }{\partial m^2} \int _{m^2}^\infty dm_0^2 \ e^{-m_0^2 s} J_0\left[ L \sqrt{m_0^2-m^2}\right] \nonumber \\&= -\frac{\partial }{\partial m^2} \int _{m^2}^\infty dm_0^2 \ J_0\left[ L \sqrt{m_0^2-m^2}\right] \int _0^\infty ds \ K_0(s) e^{-m^2_0 s}\nonumber \\&= - \frac{\partial }{\partial m^2} \int _{m^2}^\infty dm_0^2 \ J_0\left[ L \sqrt{m_0^2-m^2}\right] G_{\mathrm{std}} (m_0) \end{aligned}$$

(30)

In arriving at the last equality I have used the fact that the standard QFT propagator (without quantum corrections) corresponding to a mass $m_0$ in this space is given by the integral

$$\begin{aligned} G_{\mathrm{std}}(m_0) = \int _0^\infty ds \ K_0(s) e^{-m_0^2 s} \end{aligned}$$

(31)

Equation (30) relates the quantum corrected propagator for mass m to the standard QFT propagator for mass $m_0$ in an arbitrary Euclidean space(time). Whenever the latter is known, the former can be computed.

Let us verify this result for the flat space(time) in which $G_\mathrm{free}$ in momentum space is given by

$$\begin{aligned} G_{\mathrm{free}}(p^2,m_0^2) = \int _0^\infty d\mu \ e^{-\mu (p^2 + m_0^2)} \end{aligned}$$

(32)

Using this expression in Eq. (30), changing variable to $x^2 \equiv m_0^2 -m^2$ and carrying out the integrals, we find that

$$\begin{aligned} G_{\mathrm{QG}}(m^2)&= - \frac{\partial }{\partial m^2} \int _0^\infty 2 x \ dx\ J_0[Lx] \int _0^\infty d\mu \ e^{-\mu p^2} \, e^{-\mu (m^2 + x^2)}\nonumber \\&= - \frac{\partial }{\partial m^2} \int _0^\infty d\mu \ e^{-\mu (p^2 + m^2)} \int _0^\infty 2 x \ dx\ J_0[Lx]e^{-\mu x^2} \nonumber \\&= \int _0^\infty d\mu \ e^{-\mu (p^2 + m^2)- (L^2/4\mu )} \end{aligned}$$

(33)

where, to obtain the last equality, I have used the identity in Eq. (25). Clearly, Eq. (33) gives the correct quantum gravity corrected propagator in flat space(time).

Euclidean path measure

Let us start with the definition of path measure for ${\mathcal {G}}\equiv mG$ through the equation

$$\begin{aligned} {\mathcal {G}} \equiv m G \equiv \int _0^\infty d\ell \ N(\ell )\, e^{-m\ell } \end{aligned}$$

(34)

Very often we will work with ${\mathcal {G}}$ rather than G. In Fourier space, ${\mathcal {G}}$ has the dimensions of length thereby making $N(\ell )$ in Eq. (34) dimensionless in Fourier space. On Fourier transforming the corresponding path measure in real space acquires the dimension of $L^{-D}$, which allows it to be interpreted as a spacetime density. Of course, it is assumed that N is independent of m; that is, $N(\ell )$ is treated as the inverse Laplace transform of the rescaled propagator ${\mathcal {G}}$ from the variable m to variable $\ell$.

This definition can be illustrated in the standard free field case taking both the propagator and path measure in the Fourier space. In that case one can easily verify that

$$\begin{aligned} {\mathcal {G}}_{\mathrm{free}} (p^2,m) \equiv m G_{\mathrm{free}} (p^2,m) = \frac{m}{m^2+p^2} = \int _0^\infty d\ell \ N_{free}(\ell ) \, e^{-m\ell } \end{aligned}$$

(35)

holds with the following choice for $N(\ell )$

$$\begin{aligned} N_{free}(\ell ) = \cos p\ell \end{aligned}$$

(36)

which is indeed the inverse Laplace transform. (One can also satisfy Eq. (35), treated purely as an integral relation by the choice $N(\ell )=e^{-(p^2/m)\ell }$; but this is not acceptable since we want $N(\ell )$ to be independent of m. This is why I define N as the inverse Laplace transform of ${\mathcal {G}}$.)

One can also express the quantum corrected propagator $G_{\mathrm{QG}}$ to the path measure $N_{\mathrm{std}} (\ell )$ in an arbitrary curved space(time).⁹ We begin by rewriting Eq. (30) in terms of ${\mathcal {G}}_{\mathrm{std}}$ in the integrand, getting

$$\begin{aligned} G_{\mathrm{QG}}(m^2) = - \frac{\partial }{\partial m^2} \int _{m^2}^\infty dm_0^2\ \frac{J_0}{m_0} \, {\mathcal {G}}_{\mathrm{std}}(m_0) \end{aligned}$$

(37)

Expressing ${\mathcal {G}}_{\mathrm{std}}$ in terms of $N_{\mathrm{std}}$ using Eq. (35) and multiplying both sides of Eq. (37) by m we get the result

$$\begin{aligned} {\mathcal {G}}_{\mathrm{QG}}&= - m \frac{\partial }{\partial m^2} \int _{m^2}^\infty 2 dm_0\ \int _0^\infty d\ell \ N_{\mathrm{std}}(\ell )\, e^{-m_0\ell } J_0 \nonumber \\&= - \frac{\partial }{\partial m} \int _0^\infty d\ell \ N_\mathrm{std}\int _{m^2}^\infty dm_0\ e^{-m_0 \ell } J_0 \left[ L\sqrt{m_0^2 - m^2}\right] \nonumber \\&= - \frac{\partial }{\partial m}\int _0^\infty d\ell \ N_\mathrm{std}(\ell )\int _1^\infty dx\ m e^{-mx\ell } J_0\left[ mL \sqrt{x^2 -1}\right] \end{aligned}$$

(38)

The integral can be evaluated using the identity

$$\begin{aligned} \int _1^\infty dx\ e^{-\alpha x} \, J_0(\beta \sqrt{x^2 -1}) = \frac{e^{-\sqrt{\alpha ^2 + \beta ^2}}}{\sqrt{\alpha ^2+\beta ^2}} \end{aligned}$$

(39)

to give the rather nice result

$$\begin{aligned} {\mathcal {G}}_{\mathrm{QG}} (x_1,x_2; m) = \int _0^\infty d\ell \ N_\mathrm{std}(x_1,x_2; \ell ) \exp \left( - m \sqrt{\ell ^2+L^2}\right) \end{aligned}$$

(40)

This result tells you that one can interpret the quantum correction involving the zero point length as a simple replacement: $\ell \rightarrow \sqrt{\ell ^2 + L^2}$ without changing the path measure at all! This is a viable (though not unique) interpretation.

An alternative interpretation is to keep the path integral amplitude to be the same (i.e as $\exp -[m\sigma (x,x')]$) but introduce the quantum gravity corrections on the path measure changing $N_{std}$ to $N_{QG}$. To do this, we will use the relation in Eq. (34) between path measure and the propagator. If we use Eq. (34) with $N (\ell ) = N_{\mathrm{std}} (\ell )$ we get the standard QFT result ${\mathcal {G}}_{\mathrm{std}}(m) = m G_{\mathrm{std}} (m^2)$; on the other hand, if we use Eq. (34) with a suitable $N (\ell ) = N_{\mathrm{QG}} (\ell )$ we should get the quantum corrected propagator ${\mathcal {G}}_\mathrm{QG}(m)= m G_{\mathrm{QG}} (m^2)$. (Here, as everywhere else, I do not explicitly display the space(time) dependencies; to be precise, ${\mathcal {G}}(m)={\mathcal {G}}(x_1,x_2;m)$ and $N (\ell )=N (x_1,x_2;\ell )$.) It is easy to determine $N_{\mathrm{QG}} (\ell )$ by changing the integration variable in Eq. (40) from $\ell$ to $\mu$ through $\mu ^2 = \ell ^2 + L^2$ and rewrite Eq. (40) in the form:

$$\begin{aligned} {\mathcal {G}}_{\mathrm{QG}} (x_1,x_2; m)= \int _0^\infty \frac{\mu \, d\mu }{\sqrt{\mu ^2-L^2}} \, \theta (\mu - L) \, N_{\mathrm{std}} (x_1,x_2; \ell = \sqrt{\mu ^2 - L^2} ) \, e^{-m\mu } \nonumber \\ \end{aligned}$$

(41)

In this form we keep the path integral factor to the standard one $\exp -m\ell$ but change the path measure. The quantum corrected path measure $N_{\mathrm{QG}}(\mu )$ is related to the standard QFT path measure $N_{\mathrm{std}}(\ell )$ by the simple relation

$$\begin{aligned} N_{\mathrm{QG}}(x_1,x_2; \mu ) = \frac{\mu }{\sqrt{\mu ^2 - L^2}}\, N_\mathrm{std}\left[ x_1,x_2; \ell = \sqrt{\mu ^2-L^2}\right] \theta (\mu - L) \end{aligned}$$

(42)

In this interpretation, the path measure for lengths $\mu$ less than the zero point length L are irrelevant to physics and does not contribute. For $\mu > L$, a simple rescaling takes care of the change from $N_{\mathrm{std}}$ to $N_{\mathrm{QG}}$. It should be stressed that the whole interpretation depends on $N_{\mathrm{QG}}$ being a purely geometrical construct that is independent of the mass m of the field, which is clearly seen in the above expression.

In flat space(time) we can easily verify this result. It is convenient to use the momentum space expressions for the propagators as well as for $N_{\mathrm{free}}$ and $N_{\mathrm{QG}}$ for this purpose. In flat space(time) we have the result (in momentum space) given by the simple expression $N_{\mathrm{free}}(p, \ell ) = \cos (p\ell )$. Therefore, Eq. (42) gives the corresponding $N_{\mathrm{QG}}(p,\ell )$ to be:

$$\begin{aligned} N_{\mathrm{QG}}(p, \ell ) = \theta (\ell - L) \frac{\ell \cos p\sqrt{\ell ^2- L^2}}{\sqrt{\ell ^2-L^2}} \end{aligned}$$

(43)

So ${\mathcal {G}}_{\mathrm{QG}}(p)$ is given by the integrals in either Eq. (40) or Eq. (41). Using Eq. (40) we get:

$$\begin{aligned} {\mathcal {G}}_{\mathrm{QG}} = m\, G_{\mathrm{QG}}(p, L) = \int _0^\infty d\nu \ e^{-m \sqrt{L^2+\nu ^2}} \ \cos (p\nu ) = \frac{mL}{\sqrt{p^2+ m^2}} K_1\left[ L(p^2+m^2)^{1/2}\right] \nonumber \\ \end{aligned}$$

(44)

which is, of course, the standard result. To arrive at the final result we have used the cosine transform:

$$\begin{aligned} \int _0^\infty dx (\cos bx)\, e^{-\beta \sqrt{\gamma ^2 + x^2}} = \frac{\beta \gamma }{\sqrt{\beta ^2 + b^2}} \, K_1 \left[ \gamma \sqrt{\beta ^2 + b^2}\right] \end{aligned}$$

(45)

Finally, let us compute $N_{\mathrm{std}}(\ell )$ in real space by a Fourier transform of Eq. (43). This will lead to the integral:

$$\begin{aligned} N_{\mathrm{QG}}(\ell ,x) = \frac{\ell \, \theta (\ell -L)}{\sqrt{\ell ^2 - L^2}} \, \int \frac{d^D p}{(2\pi )^D} \, e^{ip \cdot x} \ \cos p\sqrt{\ell ^2 - L^2} \end{aligned}$$

(46)

To evaluate this expression we need a standard result. If

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

(47)

then we can write

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

(48)

Using Eq. (48) and the cosine transform we can compute this integral and obtain

$$\begin{aligned} I = \int \frac{d^Dp}{(2\pi )^D}\ e^{ip\cdot x} \cos pR = \frac{\theta (R^2 - x^2)}{\pi ^{(D-1)/2}} \, \frac{1}{\Gamma \left( - \frac{D-1}{2}\right) } \, \frac{R}{(R^2-x^2)^{(D+1)/2}} \end{aligned}$$

(49)

This leads to the result

$$\begin{aligned} N_{\mathrm{QG}}(\ell ,x) = C(D)\ \theta \left[ \ell ^2 -(x^2 + L^2)\right] \, \frac{\ell }{ \left[ \ell ^2 -(x^2 + L^2)\right] ^{(D+1)/2}} \end{aligned}$$

(50)

where

$$\begin{aligned} C(D) = \frac{1}{\pi ^{(D-1)/2}}\, \frac{1}{\Gamma \left( - \frac{D-1}{2}\right) } \end{aligned}$$

(51)

In $D=4$, this leads to the result

$$\begin{aligned} N_{\mathrm{QG}}^{D=4}(\ell ,x) = \frac{3}{4\pi ^2}\ \theta \left[ \ell ^2 - (x^2 + L^2)\right] \, \frac{\ell }{\left[ \ell ^2 - (x^2 + L^2)\right] ^{5/2}} \end{aligned}$$

(52)

The result without zero point length $N_{\mathrm{std}}$ can, of course, be obtained by putting $L =0$ in these expressions. Note that only paths which contribute are those with a length $\ell ^2 > x^2 + L^2$. The singularity structure at $\ell ^2 = x^2 + L^2$ should be handled by differentiating the expressions with respect to $L^2$ twice.