Euler-Heisenberg Lagrangian and photon circular polarization
aa r X i v : . [ h e p - ph ] O c t Euler-Heisenberg Lagrangian and photon circularpolarization
Iman Motie a and She-Sheng Xue b a) Department of Physics, Mashhad Branch, Islamic Azad University, Iranb) ICRANet, P.zza della Repubblica 10, I-65122 Pescara, & Physics Department,University of Rome “La Sapienza”, Italy Abstract
Considering the effective Euler-Heisenberg Lagrangian, i.e., non-linear photon-photon interactions, we study the circular polarization of electromagnetic radia-tion based on the time-evolution of Stokes parameters. To the leading order, wesolve the Quantum Boltzmann Equation for the density matrix describing an en-semble of photons in the space of energy-momentum and polarization states, andcalculate the intensity of circular polarizations. Applying these results to a linearpolarized thermal radiation, we calculate the circular polarization intensity, anddiscuss its possible relevance to the circular polarization intensity of the CosmicMicrowave Background radiation.
PACS: 73.50.Fq, 42.50.Xa, 98.70.Vc
Introduction.
Modern cosmological observations of the cosmic microwave background(CMB) radiation provide important evidences to understand our Universe. Cosmolog-ical informations encoded in the CMB radiation concerns not only temperature fluc-tuations and the spectrum of anisotropy pattern, but also the intensity and spectrum e-mail: [email protected] and [email protected]
1f linear and circular polarizations. It is generally expected that some relevant lin-ear and circular polarizations of CMB radiation should be present, and polarizationfluctuations are smaller than temperature fluctuations [1]. Recently, there are severalongoing experiments [2] to attempt to measure CMB polarizations. Theoretical studiesof CMB polarizations were carried out in Refs. [3, 4], and numerical calculations [5, 6]have confirmed that about 10% of the CMB radiation fields are linear polarizations,via the Compton and Thompson scatterings of unpolarized photons at the last scatter-ing surface (the redshift z ∼ ). It is important from theoretical points of view tounderstand the generation of CMB linear and circular polarizations.In principle, under effects of background fields, particle scatterings and tempera-ture fluctuations, linear polarizations of CMB radiation field propagating from the lastscattering surface can rotate each other and convert to circular polarizations. This isdescribed by the formalism of Faraday rotation (FR) and conversion (FC) [7], and theconversion from linear to circular polarizations is given by the time evolution of theStokes parameter V : ˙ V = 2 U ddt (∆ φ F C ) , (1)where ∆ φ F C is the Faraday conversion phase shift [8].Refs. [8, 9] present the role of background magnetic fields in producing the CMBcircular polarization, and ∆ φ F C ∼ − for micro gauss magnetic fields. In refs. [10, 11],the angular power spectrum of CMB circular polarizations and relevant correlations arestudied in the case that circular polarization is generated by photon-electron scatteringin the presence of magnetic fields. I is shown that Lorentz symmetry violation in someextension of standard model for particle physics [9, 12] and an axion-like cosmologicalpseudoscalar field [13] can generate the CMB circular polarization. Ref. [9] showsthat the noncommutative QED with the Seiberg-Witten expansion of fields in the lastscattering surface can also generate the CMB circular polarization, and the FC phaseshift ∆ φ F C ∼ − .In this letter we show that circular polarizations of radiation fields can be generatedfrom the effective Euler-Heisenberg Lagrangian (see review articles in refs. [14, 15, 16]).By taking into account this effective Lagrangian, and using the Quantum BoltzmannEquation [4], we study the time-evolution of the Stokes parameter V . Applying ourresults to the homogeneous CMB radiation with non-vanishing linear polarizations, weobtain the upper limit of circular polarization intensity V /T < × − in unitsof thermal temperature T of the CMB radiation, and the corresponding ∆ φ F C < . × − . 2 tokes parameters. A nearly monochromatic electromagnetic wave propagating inthe ˆ z -direction is described by: E x = a x ( t ) cos[ ω t − θ x ( t )] , E y = a y ( t ) cos[ ω t − θ y ( t )] , (2)where amplitudes a x,y and phase angles θ x,y are slowly varying functions with respect tothe period T = 2 π/ω . Any correlation between the a x - and a y -components indicatespolarizations of electromagnetic waves. In a classical description [17], Stokes param-eters, which describe polarization states of a nearly monochromatic electromagneticwave, are defined as the following time averages: I c = h a x i + h a y i ,Q c = h a x i − h a y i ,U c = h a x a y cos( θ x − θ y ) i ,V c = h a x a y sin( θ x − θ y ) i , (3)where the parameter I c is total intensity, Q c and U c intensities of linear polarizations ofelectromagnetic waves, whereas the V c parameter indicates the difference between left-and right- circular polarizations intensities. Linear polarization can also be character-ized through a vector of modulus P L ≡ p Q c + U c . It is important to notice that theStocks parameters (3) are defined for a monochromatic electromagnetic wave with adefinite momentum k . Given a linear polarization, one can always transform them toa coordinate system where Q c or U c vanishes leaving no circular polarization V c = 0.In order to generate a net circular polarization via birefringence there must be somespecial coordinate system so that one linear polarization state propagates differentlyfrom the other due to interactions.In a quantum-mechanical description, Stokes parameters can be equivalently definedas follows. An arbitrary polarized state of a photon ( | k | = | k | ), propagating in theˆ z -direction, is given by | ǫ i = a exp( iθ ) | ǫ i + a exp( iθ ) | ǫ i , (4)where linear bases | ǫ i and | ǫ i indicate the polarization states in the x - and y -directions.Quantum-mechanical operators in this linear bases, corresponding to Stokes parameter,are given by ˆ I = | ǫ ih ǫ | + | ǫ ih ǫ | , ˆ Q = | ǫ ih ǫ | − | ǫ ih ǫ | , ˆ U = | ǫ ih ǫ | + | ǫ ih ǫ | , ˆ V = i | ǫ ih ǫ | − i | ǫ ih ǫ | . (5)3n ensemble of photons in a general mixed state is described by a normalized den-sity matrix ρ ij ≡ ( | ǫ i ih ǫ j | / tr ρ ), and the dimensionless expectation values for Stokesparameters are given by I ≡ h ˆ I i = tr ρ ˆ I = 1 , (6) Q ≡ h ˆ Q i = tr ρ ˆ Q = ρ − ρ , (7) U ≡ h ˆ U i = tr ρ ˆ U = ρ + ρ , (8) V ≡ h ˆ V i = tr ρ ˆ V = iρ − iρ , (9)where “tr” indicates the trace in the space of polarization states. This shows therelationship between four Stokes parameters and the 2 × ρ for photonpolarization states. Euler-Heisenberg Lagrangian and circular polarizations.
The Euler-Hesinbergeffective Lagrangian is given as follows: £ eff = £ + δ £ , (10)where the first term £ = − F µν F µν is the classical Maxwell Lagrangian, and thesecond term δ £ δ £ ≈ α m (cid:20) ( F µν F µν ) + 74 ( F µν ˜ F µν ) (cid:21) , (11)where m is the electron mass, ˜ F µν = ǫ µναβ F αβ (see review articles in refs. [14, 15, 16]).We express the electromagnetic field strength F µν = ∂ µ A ν − ∂ ν A µ , and free gaugefield A µ in terms of plane wave solutions in the Coulomb gauge [18], A µ ( x ) = Z d k (2 π ) k h a i ( k ) ǫ iµ ( k ) e − ik · x + a † i ( k ) ǫ ∗ iµ ( k ) e ik · x i , (12)where ǫ iµ ( k ) are the polarization four-vectors and the index i = 1 ,
2, representing twotransverse polarizations of a free photon with four-momentum k and k = | k | . a i ( k )[ a † i ( k )] are the creation [annihilation] operators, which satisfy the canonical commuta-tion relation h a i ( k ) , a † j ( k ′ ) i = (2 π ) k δ ij δ (3) ( k − k ′ ) . (13)The density operator describing an ensemble of free photons in the space of energy-momentum and polarization state is given byˆ ρ = 1tr( ˆ ρ ) Z d p (2 π ) ρ ij ( p ) a † i ( p ) a j ( p ) , (14)4here ρ ij ( p ) is the general density-matrix (6-9) in the space of polarization states witha fixed energy-momentum “ p ”. The number operator D ij ( k ) ≡ a † i ( k ) a j ( k ) and itsexpectation value is defined by h D ij ( k ) i ≡ tr[ ˆ ρD ij ( k )] = (2 π ) δ (0)(2 k ) ρ ij ( k ) . (15)The time evolution of photon polarization states is related to the time evolution of thedensity matrix ρ ij ( k ), which is governed by the following Quantum Boltzmann Equation(QBE) [4],(2 π ) δ (0)(2 k ) ddt ρ ij ( k ) = i h (cid:2) H I ( t ); D ij ( k ) (cid:3) i − Z dt h (cid:2) H I ( t ); (cid:2) H I (0); D ij ( k ) (cid:3)(cid:3) i , (16)where the interacting Hamiltonian H I ( t ) = − δ £ (11). The first term on the right-handed side is a forward scattering term, and the second one is a higher order collisionterm.It is known that the linear Maxwell Lagrangian £ in Eq. (10) does not gener-ate circular polarizations. We attempt to compute the effect of the non-linear EulerHeisenberg Lagrangian (11) on the generation of circular polarizations by using QBE(16). Eq. (11) is perturbatively small, at the order of α , so that we only computethe first order of QBE, i.e. the first term in r.h.s. of Eq. (16), and neglect the secondterm which is of the order of α . The contribution from the first term ( F µν F µν ) inEq. (11) vanishes, because it is a squared Maxwell action and commutates with thenumber operator D ij . While the second nonlinear term ( F µν ˜ F µν ) does not commutatewith the number operator D ij and gives non-vanishing contributions.As a result, we approximately obtain the time-evolution equation for the densitymatrix,(2 π ) δ (0)2 k ddt ρ ij ( k ) ≈ i h (cid:2) H I ( t ) , D ij ( k ) (cid:3) i = 56 α m (2 π ) δ (0) ǫ µναβ ǫ σν ′ γβ ′ k γ k µ [ ǫ sν ( k ) ǫ l ′ β ′ ( k )] × h ρ l ′ j ( k ) δ si − ρ is ( k ) δ l ′ j + ρ sj ( k ) δ l ′ i − ρ il ′ ( k ) δ sj i × Z d p (2 π ) p p α p σ [ ǫ s ′ β ( p ) ǫ lν ′ ( p )][ ρ ls ′ ( p ) + ρ s ′ l ( p ) + δ s ′ l ] . (17)The calculations are tedious, but straightforward. We first apply the Wick theoremto arrange all creation operators to the left and all annihilation operators to the right,then we use the contraction rule h a † s ′ ( p ′ ) a s ( p ) i = 2 p (2 π ) δ ( p − p ′ ) ρ ss ′ ( p ) , (18)5o calculate all possible contractions of creation and annihilation operators a † i and a j .For example, we calculate the expectation value h p | a † s ′ ( p ′ ) a s ( p ) a † l ′ ( q ′ ) a l ( q ) | p i = h p | a † s ′ ( p ′ ) a † l ′ ( q ′ ) a s ( p ) a l ( q ) | p i + 2 p (2 π ) δ sl ′ δ ( p − q ′ ) h p | a † s ′ ( p ′ ) a l ( q ) | p i = 4 p q (2 π ) δ ( p − p ′ ) δ ( q − q ′ ) ρ ss ′ ( p ) ρ ll ′ ( q )+ 4 p q (2 π ) δ ( p − q ′ ) δ ( q − p ′ ) ρ s ′ l ( q )[ δ sl ′ + ρ sl ′ ( p )] , where the first line results from the Wick theorem and commutation relations (13),while the second line results from all possible contractions (18) of operators a † s and a s .Using Eq. (17), we obtain the time-evolutions for Stocks parameters (6-9) as follows:˙ I ( k ) = 0 , (19)˙ Q ( k ) = ˆ X n [ ρ ( k ) − ρ ( k )][ ρ ( p ) − ρ ( p )] × [ ǫ ν ( k ) ǫ β ′ ( k ) ǫ β ( p ) ǫ ν ′ ( p ) + ǫ ν ( k ) ǫ β ′ ( k ) ǫ β ( p ) ǫ ν ′ ( p )] o = − ˆ X n V ( k ) V ( p )[ ǫ ν ( k ) ǫ β ′ ( k ) ǫ β ( p ) ǫ ν ′ ( p ) + ǫ ν ( k ) ǫ β ′ ( k ) ǫ β ( p ) ǫ ν ′ ( p )] o , (20)˙ U ( k ) = ˆ X n [ ρ ( k ) − ρ ( k )][ ρ ( p ) − ρ ( p )] ǫ ν ( k ) ǫ β ′ ( k ) ǫ β ( p ) ǫ ν ′ ( p ) o = ˆ X n iV ( k ) Q ( p ) ǫ ν ( k ) ǫ β ′ ( k ) ǫ β ( p ) ǫ ν ′ ( p ) o , (21)˙ V ( k ) = ˆ X n [ ρ ( k ) − ρ ( k )][ ρ ( p ) + ρ ( p )] ǫ ν ( k ) ǫ β ′ ( k ) ǫ β ( p ) ǫ ν ′ ( p )+ [ ρ ( k ) − ρ ( k )][ ρ ( p ) − ρ ( p )] ǫ ν ( k ) ǫ β ′ ( k ) ǫ β ( p ) ǫ ν ′ ( p ) o = ˆ X n Q ( k ) U ( p ) ǫ ν ( k ) ǫ β ′ ( k ) ǫ β ( p ) ǫ ν ′ ( p )+ iV ( k ) Q ( p ) ǫ ν ( k ) ǫ β ′ ( k ) ǫ β ( p ) ǫ ν ′ ( p ) o , (22)where k indicates the energy-momentum state of incoming photons in a radiation field,and p indicates the energy-momentum states of virtual photons in vacuum, and theoperator ˆ X is defined as following integral overall energy-momentum states p ,ˆ X n · · · o ≡ × α m k Z d p (2 π ) p h ǫ µναβ ǫ σν ′ γβ ′ k γ k µ p α p σ i n · · · o . (23)The I modes represent the ensemble of photons. The Q and U modes represent theensemble of linearly polarized photons, and the V mode represents the ensemble ofcircularly polarized photons.Eq. (11) gives an interacting vortex of four photons. Eqs. (19-22) result from thetadpole diagram of a photon loop integrating all contributions “ p ” of virtual photons6n vacuum (see Eq. 23). This indicates that polarization states of a propagating photonwith momentum “ k ” interact with those of virtual photons in vacuum. If the photon“ k ” is not initially polarized, i.e., Q ( k ) = U ( k ) = V ( k ) = 0, then Eqs. (19-22) show thatthe photon “ k ” propagating through vacuum does not acquire polarizations. Instead,the photon “ k ” is linearly polarized, as if there were a particular orientation of localmagnetic field. Interacting with this local magnetic field, virtual photons can developcircular polarization states, in turn these states back-react with polarization states ofthe photon “ k ”. As a result, Eqs. (19-22) show that the photon “ k ” acquires a netcircular polarization.It is important to notice that in the right-handed side of Eq. (22), the linearlypolarized modes Q ( k ) and U ( p ) are in different momentum states “ k ” and “ p ” so thatthey are independent modes. This cannot be made by a coordinate transformationso that one of them ( U or V ) vanishes, as discussed for the Stocks parameters (3) or(6-9) for a monochromatic electromagnetic wave with a definite momentum “ k ”. Inaddition, the right-handed sides of Eqs. (20,21,22) show nonlinear interactions between Q , U and V modes of a given momentum state “ k ” of a radiation field and all possiblemomentum states “ p ” from vacuum contributions. In these nonlinear interactions, Q , U and V modes differently interact with each other leading to circular polarizations,i.e., non-vanishing V ( k ) modes, provided Q ( k ) and U ( p ) are non zero.In Eq. (19), ˙ I = 0 implies in the ensemble of photons, the total intensity of pho-tons is constant in time-evolution. In Eqs. (20,21) and (22), the time-evolution ˙ Q ,˙ U and ˙ V are given by the combinations of Q , U and V modes, which indicates arotation or conversion between these modes as long as the effective interaction (11)acts. The time-evolution ˙ V is proportional to Q and U modes. This indicates thatan ensemble of linearly polarized photons will acquire circular polarizations due to theEuler-Heisenberg Lagrangian (11). We are interested in considering an initially linearpolarized electromagnetic radiation propagating through vacuum, and calculating howmuch the intensity of circular polarization can be converted from the intensity of linearpolarization, due to the non-linear Euler-Heisenberg interaction. Intensity of circular polarizations in CMB.
We apply our results (19-22) to theensemble of thermal CMB photons, f BB ( p ) = 1 / ( e p/T − T and photon momentum p are in the comoving frame. Thus photon energy and numberdensities are given by: ε γ = 2 Z d p (2 π ) p f BB ( p ) = π T , n γ = 2 Z d p (2 π ) f BB ( p ) = 2 ζ (3) π T . (24)7nd the mean energy for each thermal photon ε γ n γ = π T [30 ζ (3)] ≈ . T, (25)corresponds to the intensity I c (3). We consider that the thermal radiation is initiallypolarized and propagates through vacuum. In order to calculate the final intensity ofcircular polarizations (22), we approximate Q ( k ) ǫ β ′ ( k ) ǫ ν ( k ) ≈ C Q f BB ( k ) δ βν ′ , U ( p ) ǫ β ( p ) ǫ ν ′ ( p ) ≈ C U f BB ( p ) δ βν ′ , (26)where coefficient C Q ( C U ) is the ratio of linear Q ( U )-polarization intensity and totalintensity. The coefficient C Q represents the linear polarization of the real photon “ k ”propagating in vacuum C Q <
1, while C U represents the sum over all contributions oflinear polarization of virtual photons “ p ” in vacuum, and C U ≃ C Q and C U areindependent of each other, because they are associated to different momentum states k and p of photons.Assuming that the converted intensity of circular polarization is much smaller thanthe intensity of linear polarization, we neglect the second term in the right-handedside of Eq. (22). Integrating Eq. (22) over all momentum states p [see Eq. (23)], weapproximately obtain ˙ V ( k ) ∼ = (cid:16) π α (cid:17)(cid:16) Tm (cid:17) C U C Q [ kf BB ( k )] , (27)where k is the momentum of thermal photons. The form of expression (27) can beunderstood as the rate of converting the linearly polarized mode to a circularly po-larized mode. The linearly polarized Q -mode of momentum state k interacts withlinearly polarized U -modes of all momentum states p in vacuum, and converts to thecircularly polarized V -mode of momentum state k . The factor [ kf BB ( k )] is due to theenergy-spectrum of thermal photons and ( T /m ) comes from the summation over allmomentum states p (24). Integrating Eq. (27) overall energy-momentum states “ k ” ofthe black-body distribution f BB ( k ), and normalizing it by the total number-density n γ of thermal photons, we obtain dVdt ∼ = (cid:16) π α (cid:17)(cid:16) Tm (cid:17) (cid:16) ε γ n γ (cid:17) C U C Q , V ≡ n γ Z d k (2 π ) V ( k ) . (28)Finally, multiplying the mean intensity ε γ /n γ , we obtain dVdt ∼ = (cid:16) π α (cid:17)(cid:16) Tm (cid:17) (cid:16) ε γ n γ (cid:17) C U C Q ; (29)8nalogously, we redefine the Stokes parameters Q and U in energy units by multiplyingthe mean energy ε γ /n γ , U = (cid:16) ε γ n γ (cid:17) n γ Z d p (2 π ) U ( p ) ≈ C U (cid:16) ε γ n γ (cid:17) , (30)and U ⇒ Q , corresponding to those in Eq. (5).To estimate the V , we integrate over the comoving time R dt = R dz/H ( z ), wherethe redshift z ∈ [0 , ], the Hubble function H ( z ) = H [Ω M ( z + 1) + Ω Λ )] / forΩ M ≃ .
3, Ω Λ ≃ . H = 75 km/s/Mpc, and the temperature T = T (1 + z )[ T ≈ . K ◦ = 2 . × − eV = (0 . − ] in the standard cosmology [19, 20].From Eqs. (25,29) we obtain in units of the present CMB temperature T ,∆ VT ≃ . (cid:16) π α (cid:17)(cid:16) T m (cid:17) T C U C Q Z dz (1 + z ) /H ( z ) ≈ × − C U C Q < × − . (31)We find that this intensity of circular polarizations is very small, as compared with theCMB anisotropy ∆ T /T ≈ − [21], which measures the inhomogeneity of the CMBradiation. From Eqs. (1), (29) and (30) , and we obtain the Faraday conversion phaseshift ∆ φ F C ≃ . (cid:16) π α (cid:17)(cid:16) T m (cid:17) T C Q Z dz (1 + z ) /H ( z ) ≈ . × − C Q < . × − . (32)Our results (31,32) give the upper limit of the effects of non-linear Euler-HeisenbergLagrangian on the intensity of CMB circular polarization.Analogously to Ref. [22], which also discusses the effect of photon-photon scatteringson the CMB circular polarization, we take the linear polarization coefficient C Q ≈ × − at the last scattering surface, corresponding to ∆ T Q /T with the maximum∆ T Q ≈ × − K ◦ in the WMAP data [23], we obtain ∆ V /T ≈ . × − , and∆ φ F C ≈ . × − . This result implies that the Euler-Heisenberg effect on the CMBcircular polarization could be rather important, compared with those effects mentionedin the introductory paragraph. Conclusion and remarks.
In this letter, by approximately solving the first order ofQuantum Boltzmann Equation for the density matrix of a photon ensemble, and time-evolution of Stokes parameters, we show that propagating photons convert their linearpolarizations to circular polarizations by the nonlinear Euler-Heisenberg interactions.9e discussed this Euler-Heisenberg effect on the circular polarization of CMB photons,and showed that this effect is very small, as compared with the present CMB temper-ature T . Nevertheless, observational studies on such circular polarization are clearlywarranted. What and how do we need to observe? Is non-zero circular polarizationat one point in the sky enough, or should there be correlations with the pattern oflinear polarization? To our knowledge, there is not any current polarization experimentthat directly measures the CMB circular polarization. However, in the next five yearsconsiderably more detailed information about the CMB polarization will be deliveredby the Planck satellite [24] and ground based, high resolution polarization experimentssuch as ACTPol [25], PIXIE [26], SPIDER [27], PolarBear [28], and SPTPol [29]. Thesensitivity (polarization) ∆ T Q /T of the Planck satellite for hight and low frequencyis in the order of 10 − . This seems to be still far from the CMB circular polarization∆ V /T ≈ . × − estimated in this letter. On the other hand, it would be interestingto see this Euler-Heisenberg effect on the circular polarization of laser photons [30]. Acknowledgment.
One of the authors, I. Motie thanks Professor R. Ruffini for hishospitality, during his stay at ICRANet Pescara, Italy, where this work is done. Wethank the anonymous referee for his comments and suggestions.