Gravity and handedness of photons

TL;DR

This paper explores how strong gravitational fields can induce a quantum optical activity in electromagnetic radiation, causing polarization changes over time due to a quantum anomaly related to electric-magnetic duality.

Contribution

It introduces the concept that spacetime geometry can act as an optically active medium for photons, revealing a quantum effect linked to duality anomalies in Maxwell theory.

Findings

Spacetime geometry influences photon polarization via quantum effects.

Polarization states evolve over time without electromagnetic sources.

The effect is rooted in a duality anomaly in Maxwell's equations.

Abstract

Vacuum fluctuations of quantum fields are altered in presence of a strong gravitational background, with important physical consequences. We argue that a non-trivial spacetime geometry can act as an optically active medium for quantum electromagnetic radiation, in such a way that the state of polarization of radiation changes in time, even in the absence of electromagnetic sources. This is a quantum effect, and is a consequence of an anomaly related to the classical invariance under electric-magnetic duality rotations in Maxwell theory.

Equations

$$⟨{\partial }_{\mu }{j}_5^{\mu }⟩=-\frac{q^2\hslash }{8{\pi }^2}{F}_{\mu \nu }{{}^{\star }F}^{\mu \nu }=-\frac{q^2\hslash }{2{\pi }^2}{E}_{\mu }{B}^{\mu },$$

$$j_D^{\mu }=\frac{1}{2}\left(A_{\nu }{}^{\star }{F}^{\mu \nu }-F^{\mu \nu }{Z}_{\nu }\right).$$

$$\vec{\nabla }\times {\vec{A}}_{\pm }=\pm i{\partial }_t{\vec{A}}_{\pm },\vec{\nabla }\cdot {\vec{A}}_{\pm }=0.$$

$${\beta }^a{\partial }_a\Psi (x)=0,$$

$$\displaystyle\Psi\equiv\left({\begin{array}[]{c}A_{+}^{\ i}\\ A_{-\,i}\\ \end{array}}\right)\ ,\hskip 17.07182pt\beta^{a}\equiv i\left({\begin{array}[]{cc}0&(\bar{\alpha}^{a})_{b}^{\ i}\\ -(\alpha^{a})^{b}_{\ i}&0\\ \end{array}}\right)\ .$$

$$\displaystyle\Psi\rightarrow e^{i\theta\beta_{5}}\Psi\,=\left({\begin{array}[]{c}e^{-i\theta}A_{+}^{i}\\ e^{i\theta}A_{-\,i}\\ \end{array}}\right)\ ,$$

$$\displaystyle\left<\nabla_{\mu}j_{D}^{\mu}\right>=\frac{\hbar}{24\pi^{2}}R_{\mu\nu\lambda\sigma}{{}^{\star}R}^{\mu\nu\lambda\sigma}=\frac{2\hbar}{3\pi^{2}}\,E_{\mu\nu}B^{\mu\nu},\$$

$$\Delta Q_D=\frac{2\hslash }{3{\pi }^2}{\int }_{t_1}^{t_2}{\int }_{\Sigma }{d}^{4}x\sqrt{-g}{E}_{\mu \nu }{B}^{\mu \nu }.$$

Full text

Gravity and handedness of photons

Ivan Agullo111agullo@lsu.edu

Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803-4001, USA

Adrian del Rio222adrian.rio@uv.es and Jose Navarro-Salas333jnavarro@ific.uv.es

*Departamento de Fisica Teorica and IFIC, Centro Mixto Universidad de Valencia-CSIC. Facultad de Fisica, Burjassot-46100, Valencia, Spain. *

Vacuum fluctuations of quantum fields are altered in presence of a strong gravitational background, with important physical consequences. We argue that a non-trivial spacetime geometry can act as an optically active medium for quantum electromagnetic radiation, in such a way that the state of polarization of radiation changes in time, even in the absence of electromagnetic sources. This is a quantum effect, and is a consequence of an anomaly related to the classical invariance under electric-magnetic duality rotations in Maxwell theory.

First Award in the 2017 Essay Competition of the Gravity Research Foundation.

The presence of a gravitational background introduces non-trivial effects in the propagation of electromagnetic radiation, with fundamental implications. The gravitational redshift and the deflection of light rays by massive bodies played a pivotal role in the birth of general relativity [1], and are of great importance in many areas of gravitation, cosmology, and astrophysics. It is also well-known that remarkable new features appear in the physics of quantum fields when they propagate under the influence of gravity. Renowned examples are the spontaneous creation of quanta by the expanding universe [2], the thermal radiation by black holes produced by gravitational collapse [3], and the generation of primordial density perturbations by inflation [4]. In this essay, we discuss a new effect on quantum electromagnetic fields propagating in curved spacetimes.

Our discussion resembles the study of the chiral anomaly for fermions. Recall that the theory of a zero-mass, charged spin-$1/2$ fermion satisfying the Dirac equation in Minkowski spacetime, $\gamma^{\mu}\partial_{\mu}\Psi=0$, is invariant under chiral transformations: $\Psi\rightarrow e^{i\theta\,\gamma_{5}}\Psi$. Noether’s theorem then implies that the so-called axial-vector current, $j^{\mu}_{5}=\bar{\Psi}\gamma^{\mu}\gamma_{5}\Psi$, is conserved; $\partial_{\mu}j^{\mu}_{5}=0$. This classical symmetry is present even if we let fermions interact with a classical electromagnetic field $F_{\mu\nu}$. But it is well-known that quantum fluctuations of the Dirac field lead to an anomaly [5]; the vacuum expectation value of $\partial_{\mu}j^{\mu}_{5}$ does not vanish

[TABLE]

where $q$ is the charge of the fermion, and ${}^{\star}F^{\mu\nu}=1/(2\sqrt{-g})\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$ is the dual of $F_{\mu\nu}$. In the last equality we have written the Lorentz invariant quantity $F_{\mu\nu}{{}^{\star}F}^{\mu\nu}$ in terms of the electric and magnetic parts of $F_{\mu\nu}$. This anomaly has important physical implications, e.g. it provides the mechanism to explain the pion decay to two photons $\pi^{0}\rightarrow\gamma\gamma$.

With this result in mind, we now analyze a different physical situation: the theory of a source-free electromagnetic field propagating on a classical gravitational background. The corresponding Maxwell action is invariant under duality transformations [6]: $F_{\mu\nu}\rightarrow F_{\mu\nu}\,\cos{\theta}+\,^{\star}F_{\mu\nu}\,\sin{\theta}$. In terms of the vector potential, the infinitesimal transformation reads $\delta A_{\mu}=\theta\,Z_{\mu}$, where $Z_{\mu}$, when evaluated on-shell, is the potential of the dual field ${}^{\star}F_{\mu\nu}=\nabla_{\mu}Z_{\nu}-\nabla_{\nu}Z_{\mu}$. Again, Noether’s techniques provide a conserved current $j^{\mu}_{D}$, whose expression on-shell is given by [7]

[TABLE]

$Q_{D}$, the spatial integral of the time component of $j^{\mu}_{D}$ on any hyper-surface, is a conserved quantity of the classical theory in arbitrary spacetime geometries. In Minkowski spacetime, $Q_{D}$ represents the difference in amplitude between right and left polarized components of the electromagnetic field — this is the $V$-Stokes parameter. This conservation law says that the state of polarization of light is a constant of motion.

The electromagnetic duality is indeed a remarkable symmetry. It suggested the existence of magnetic charges as a way to retain the symmetry in presence of electromagnetic sources. We argue below, however, that the gravitational interaction spoils the symmetry, even without sources, once quantum fluctuations of the electromagnetic field are taken into account.

That the transformation law for the basic variables $A_{\mu}$ looks complicated—non-local in fact—indicates that the free Maxwell theory is more naturally formulated in terms of other variables, namely self- and anti-self dual fields. We will reformulate the theory in terms of these variables, and in doing so we will see an interesting analogy with the dynamics of fermions and the associated chiral symmetry. Powerful techniques developed for fermions will then become available to study the electromagnetic duality in the quantum theory. We focus first on Minkowski spacetime, and later allow arbitrary gravitational fields.

Define the complex fields $\vec{H}_{\pm}:=\frac{1}{2}[\vec{E}\pm i\vec{B}]$. It is easy to see that they transform under duality by a simple phase, $\vec{H}_{\pm}\to e^{\mp i\theta}\vec{H}_{\pm}$. $\vec{H}_{+}$ and $\vec{H}_{-}$ are called the self- and anti-self-dual parts of the electromagnetic field. They transform under the $(1,0)$ and $(0,1)$ representations of the Lorentz group, respectively. Define the associated vector potentials $\vec{H}_{\pm}=:\pm i\,\vec{\nabla}\times\vec{A}_{\pm}$. The Maxwell equations, in the radiation gauge, read

[TABLE]

These equations can be derived either by directly writing Maxwell’s equations $dF=d\,^{\star}F=0$ in terms of $\vec{A}_{\pm}$, or as Hamilton’s equations for the Maxwell Hamiltonian using $2\vec{A}_{-}$ and $-\vec{H}_{+}$—or equivalently $2\vec{A}_{+}$ and $-\vec{H}_{-}$—as canonically conjugate variables.

The four equations (3) for $\vec{A}_{+}$ can be written more compactly as $(\alpha^{a})^{b}_{\hskip 4.26773pti}\partial_{a}A^{i}_{+}=0$, where the components of the $(\alpha^{a})^{b}_{\hskip 4.26773pti}$ matrices are easily extracted from (3)—the spacetime indices $a,b$ run from 0 to 3, and the internal index $i$ runs from 1 to 3. The equations for the anti-self-dual potential $\vec{A}_{-}$ are the complex-conjugates of the equation for $\vec{A}_{+}$. Both sets of equations can be combined in

[TABLE]

where we have defined

[TABLE]

(The bar denotes complex conjugation). The $\beta^{a}$-matrices satisfy the following (anti-) commutation properties: $\bar{\beta}^{(a}\beta^{b)}=-\eta^{ab}\,\mathbb{I}\ ,\ \bar{\beta}^{[a}\beta^{b]}=2$ diag$\left({}^{+}\Sigma^{ab},^{-}\Sigma^{ab}\right)$, where $\eta_{ab}$ is the Minkowski metric and ${}^{\pm}\Sigma^{ab}$ are the generators of the $(1,0)$ and $(0,1)$ representations of the Lorentz group, respectively. $\alpha^{a}$ and $\beta^{a}$ are the spin-1 analog of the Pauli $\sigma^{a}=(I,\vec{\sigma})$ and Dirac $\gamma^{a}$ matrices, respectively. Duality transformations are now written as

[TABLE]

where $\beta_{5}\equiv\frac{i}{16}\epsilon_{\mu\nu\sigma\rho}\bar{\beta}^{\mu}\beta^{\nu}\bar{\beta}^{\sigma}\beta^{\rho}=$ diag$\left(-\mathbb{I}_{3\times 3},\mathbb{I}_{3\times 3}\right)$.

The space of solutions of (4) is spanned by transverse monochromatic waves oscillating with positive and negative frequencies. The relation between self-duality and helicity is as follows. Self-dual monochromatic waves have (negative) positive helicity for (negative) positive frequency modes, while the opposite occurs for anti-self dual solutions. This is analog to the relation between chirality and helicity for fermions.

The generalization to curved spacetimes is straightforward. The Minkowski metric $\eta_{ab}$ and the ordinary derivative $\partial_{a}$ are replaced by the curved metric tensor $g_{\mu\nu}$ and the associated covariant derivative $\nabla_{\mu}$, and the curved spacetime $\alpha$- and the $\beta$-matrices are obtained from the flat spacetime ones by using an orthonormal tetrad or vierbein, $(\alpha^{\mu})^{\nu}_{\hskip 4.26773pti}(x)=e^{\mu}_{a}(x)\,e^{\nu}_{b}(x)\,(\alpha^{a})^{b}_{\hskip 4.26773pti}$.

We now explore the duality symmetry in the quantum theory. We will follow the functional-integral strategy used in [9] to establish the chiral anomaly of fermions. The important question is whether the measure of the path integral respects the symmetry of the action. The calculation follows steps similar to the fermionic case, after replacing all the structures associated to the $(1/2,0)$ and $(0,1/2)$ representations of the Lorentz group with the $(1,0)$ and $(0,1)$ ones, and taking care of additional subtleties arising from the gauge freedom. The reader is referred to [7] for details. In spite of the differences, the result is remarkably similar to the fermionic chiral anomaly. Namely, quantum fluctuations spoil the classical symmetry and the expectation value of the divergence of (2) —for any vacuum state— becomes

[TABLE]

where in the last equality we have written the Chern-Pontryagin topological density $R_{\mu\nu\lambda\sigma}{{}^{\star}R}^{\mu\nu\lambda\sigma}$ in terms of the electric and magnetic parts of the Weyl tensor. Compare this expression with (1). As a consequence, the duality charge $Q_{D}$ is not conserved in general spacetimes, and its change between any two instants is given by

[TABLE]

An analog of this effect arises for fermions in the creation of pairs from the vacuum by strong electric fields. In this situation the presence of a magnetic field would induce a non-zero net chirality on the particles created, as predicted by (1) [8]. Likewise, apart from gravitational tidal forces produced by $E_{\mu\nu}$, frame dragging effects, described by $B_{\mu\nu}$, are necessary to induce net polarization on the created photons.

To illustrate this, consider the process of collapse of a neutron star into a Kerr black hole. In the vicinity of the region where the event horizon will form, the gravitational field is strongly changing and spontaneous creation of photons will occur. Our result indicates that the created photons, when measured far from the star, will carry a net polarization given by $\Delta Q_{D}$. Numerical simulations indicate that for a neutron star of $M=1.73$ solar masses and angular momentum $J=0.36M^{2}$ the collapse produces around 30 photons per second more with one circular polarization than the other. This process has no classical counterpart and is different from the standard, late-time Hawking radiation, which does not contribute to $\Delta Q_{D}$.444 In fact, for an exact Kerr geometry expression (14) yields zero. Therefore, $\Delta Q_{D}$ comes from the (transient) process of collapse, in contrast to the Hawking effect, which is associated with the final, stationary black hole configuration.Although this number is small—given the short duration of the gravitational collapse—it is significant if we compare it with the $\approx 20$ total photons per second, steadily emitted by the formed black hole via Hawking radiation, with net polarization equal zero [10]. It is expected that $\Delta Q_{D}$ becomes significantly larger in more violent processes, as for instance the collision and merger of two black holes as the ones observed by LIGO [11]. But more importantly, the existence of spontaneous creation of photons implies that the stimulated counterpart must exist. Therefore, electromagnetic radiation traveling in spacetimes with a non-zero value of (14), such as the ones mentioned above, will experience a change in its net polarization. This may have implications for CMB photons; their propagation through the large scale structure would not only bend their trajectories but also affect their polarization.

Acknowledgments. We are very grateful to N. Sanchis-Gual and J. A. Font for numerical assistance in the black hole collapse, and to K. Runnels for a careful reading of the manuscript. This work was supported by the Grants No. FIS2014-57387-C3-1-P, and No. MPNS of COST Action No. MP1210, the Severo Ochoa program SEV-2014-0398 and NSF Grants No. PHY-1403943 and No. PHY-1552603. A. d. R. is supported by the Spanish FPU Ph.D. fellowship.