Revisiting equation of state for white dwarfs within finite temperature quantum field theory
RRevisiting equation of state for white dwarfs within finite temperature quantum fieldtheory
Golam Mortuza Hossain ∗ and Susobhan Mandal † Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur - 741 246, WB, India (Dated: September 25, 2019)The effects of fine-structure constant on the equation of state of degenerate matter in the whitedwarfs are computed in literature using non-relativistic considerations ab initio. Given special rel-ativity plays a key role in the white dwarf physics, such computations are therefore unsatisfactory.After reviewing the existing literature, here we employ the techniques of finite temperature rela-tivistic quantum field theory to compute the equation of state of degenerate matter in the whitedwarfs. In particular, we compute the leading order corrections due to the finite temperature andthe fine-structure constant. We show that the fine-structure constant correction remains well-definedeven in the non-relativistic regime in contrast to the existing treatment in the literature. Besides,it involves an apriori undetermined length scale characterizing the electron-nuclei interaction.
I. INTRODUCTION
In the study of degenerate matter within the whitedwarfs, as pioneered by Chandrasekhar [1, 2], the degen-erate electrons are treated as free particles which followthe Fermi-Dirac statistics. The effect of electromagneticinteraction on the degenerate matter in the white dwarfs,by the means of ‘classical’ Coulomb energy, was first con-sidered by Frenkel [3], and was followed up by Kothari[4], Auluck and Mathur [5]. However, a more accuratestudy on various corrections to the equation of state dueto the so-called Coulomb effects was done by Salpeter [6].The implications of these corrections on the mass-radiusrelation of the white dwarfs were carried out by Hamada,Salpeter [7] and Nauenberg [8].One may classify the total Coulomb effects that areconsidered by Salpeter into following broad components(see also [9, 10]). (a) The ‘classical’ Coulomb energy in-cludes the electrostatic energy of uniformly distributeddegenerate electrons within the Wigner-Seitz cells, eachsurrounding a positively charged nucleus within a rigidlattice. It includes the electron-nuclei interaction and theself-interaction of electrons. (b) The Thomas-Fermi cor-rection arises due to the radial variation of electron den-sity within a Wigner-Seitz cell. (c) The ‘exchange energy’and the ‘correlation energy’ arise due to the transverseinteractions between two electrons, essentially due to theLorentz force between them apart from its electrostaticcomponent which is already included in (a). We maymention here that the Thomas-Fermi model is a non-relativistic model and relativistic corrections to it havebeen considered for the white dwarfs in Ref. [11–13].The special relativity plays a key role in the whitedwarf physics. In particular, the existence of the Chan-drasekhar upper mass limit for the white dwarfs arises es-sentially due to the special relativity which demands thatthe physical results should be invariant under the Lorentz ∗ Electronic address: [email protected] † Electronic address: [email protected] transformations. Unfortunately, the methods employedin computing the Coulomb effects to the equation of stateof the white dwarfs use electrostatic considerations whichare non-relativistic ab initio . Therefore, from the fun-damental point of view these computations should beviewed as an approximation of the corrections that onewould expect from a Lorentz invariant computation. Fur-ther, these computations are usually performed at thezero temperature [6] (however see [14–16]). Besides, ithas been noted that the future detection of low-frequencygravitational waves from the extreme mass-ratio mergerof a black hole and a white dwarf could determine theequation of state of the degenerate matter within thewhite dwarfs with very high accuracy [17]. Such an accu-racy could probe the extent of Coulomb effects and henceprovides additional motivation to revisit the correctionsto the equation of state of the white dwarfs.In order to compute the equation of state for the whitedwarfs, a natural arena which respects Lorentz sym-metry, is provided by the finite temperature relativis-tic quantum field theory. Following the pioneering workof Matsubara [18], the techniques of finite-temperaturequantum field theory was employed in the context of quantum electrodynamics (QED) by Akhiezer and Pelet-minskii [19], and later by Freedman and McLerran [20],to compute the ground state energy of the relativisticelectron gas that includes corrections due to the fine-structure constant. However, these treatments are insuf-ficient to describe the degenerate matter in the whitedwarfs as they do not describe the dominant interac-tion, as seen in non-relativistic computations, betweenthe degenerate electrons and positively charged heaviernuclei. Therefore, in this article to describe the degen-erate matter in the white dwarfs using the frameworkof finite temperature quantum field theory, we consideran additional interaction between the electrons and pos-itively charged nuclei, described by a Lorentz invariantaction, along with the quantum electrodynamics. a r X i v : . [ a s t r o - ph . H E ] S e p II. WHITE DWARF STARS
In order to specify the associated scales, let us considerthe white dwarf star Sirius B which has observationalmass M = 1 . M (cid:12) , radius R = 0 . R (cid:12) and the effec-tive temperature T = 25922 K [21]. Therefore, its massdensity is ρ ≈ . × g/cm . In natural units that wefollow here ( i.e. speed of light c and Planck constant (cid:126) areset to unity), a fully degenerate core implies that the elec-tron density is n e ≈ . × (eV) . The correspondingFermi momentum is k F (cid:39) (3 π n e ) / ≈ . × eV.The associated temperature scale of the white dwarfs β − ≡ k B T = 2 . βk F ≈ . × , (1)which plays an important role in characterizing the whitedwarf star. A. Fermionic matter field
In order to compute the equation of state of the de-generate matter, we consider the spacetime within awhite dwarf star to be described by the Minkowski metric η µν = diag ( − , , , i.e. we ignore the corrections fromthe general relativity (as also done by Salpeter [6]). Thedegenerate electrons are fermionic degrees of freedom andare represented by the Dirac spinor field ψ along with theaction S ψ = (cid:90) d x L ψ = − (cid:90) d x √− η ψ [ iγ µ ∂ µ + m ] ψ , (2)where η = det ( η µν ) = −
1. The Dirac matrices γ µ satisfythe anti-commutation relation { γ µ , γ ν } = − η µν I . (3)The minus sign in front of η µν in the Eq. (3) is chosensuch that for given metric signature, the Dirac matricessatisfy the usual relations ( γ ) = I and ( γ k ) = − I for k = 1 , ,
3. In Dirac representation, these matrices canbe expressed as γ = (cid:18) I − I (cid:19) , γ k = (cid:18) σ k − σ k (cid:19) , (4)where Pauli matrices σ k are given by σ = (cid:18) (cid:19) , σ = (cid:18) − ii (cid:19) , σ = (cid:18) − (cid:19) . (5) B. Gauge field
The electromagnetic interaction between the fermionsare mediated by the gauge fields A µ whose free dynamics is governed by the Maxwell action S A = (cid:90) d x L A = (cid:90) d x (cid:20) − F µν F µν (cid:21) , (6)where the field strength F µν = ∂ µ A ν − ∂ ν A µ . C. Field interactions
The degenerate electrons within a white dwarf ex-perience two kinds of interactions, namely the self-interaction between the electrons and the interaction be-tween electrons and the positively charge nuclei. Theself-interaction between the electrons are mediated bygauge fields A µ and governed by the interaction term ofthe quantum electrodynamics S − I = (cid:90) d x L − I = (cid:90) d x ¯ ψ [ e γ µ A µ ] ψ , (7)where the parameter e is the dimensionless coupling con-stant.We may recall that the conserved 4-current corre-sponding to the action (2) is given by j µ = ¯ ψγ µ ψ whichrepresents the contribution of the electrons. Similarly,we may consider a background 4-current, say J µ , to rep-resent the contributions from the nuclei. Therefore, wemay model the attractive interaction between the elec-trons and the positively charged nuclei by a Lorentz in-variant action containing the current-current interaction S + I = (cid:90) d x L + I = (cid:90) d x [ − Ze d J µ ¯ ψγ µ ψ ] , (8)where the coupling constant contains the term − Ze which signifies the strength of the attractive interactionsbetween an electron and a positively charged nucleuswith atomic number Z . The parameter d is introducedin order to make the action (8) dimensionless. It has thedimension of length and and it represents an effectivelength scale associated with the current-current interac-tion.Therefore, the total action that describes the dynamicsof the degenerate electrons within a white dwarf can bewritten as S = S ψ + S A + S I = S QED + S + I , (9)where S I = S − I + S + I . The inclusion of the additionalinteraction term (8) preserves the symmetry of the ac-tion of quantum electrodynamics S QED . In other words,apart from being Lorentz invariant, the total action (9)is also invariant under local U(1) gauge transformations A µ → A µ − e ∂ µ α ( x ) and ψ ( x ) → e iα ( x ) ψ ( x ) with α ( x )being an arbitrary function. Given the coupling constant e is small, we can study the interacting theory by usingperturbative techniques. III. PARTITION FUNCTION
In a spherically symmetric star, the pressure and themass density both vary along the radial direction. Onthe other hand, in order to apply the techniques of finitetemperature quantum field theory we need to consider aspatial region which is in thermal equilibrium at a giventemperature T . Within such a region thermodynamicalquantities such as the pressure and the density are uni-form. Therefore, in order to deal with both these aspectswe consider here a finite spatial box at a given radial co-ordinate. The box is assumed to be sufficiently small sothat the pressure and the density remain uniform withinthe box yet it is sufficiently large to contain enough de-grees of freedom to achieve required thermodynamicalequilibrium. The corresponding partition function, de-scribing the degrees of freedom within the box, can beexpressed as Z = T r (cid:104) e − β ( ˆ H − µ ˆ Q ) (cid:105) , (10)where β = 1 /k B T with k B being the Boltzmann con-stant, µ refers to chemical potential and Q is the con-served charge of the system. The Hamiltonian operatorˆ H represents the matter fields described by the action(9). The trace operation is carried out over the degreesof freedom contained within the specified spatial region. A. Partition function for free fermions
The action (2) of free spinor field is invariant undera global U(1) gauge transformations ψ ( x ) → e iα ψ ( x )where α is an arbitrary constant. Consequently, thereexists an associated conserved current j µ = ¯ ψγ µ ψ suchthat ∂ µ j µ = 0. Then the corresponding conserved chargecan be expressed as Q = (cid:82) d x j ( x ) = (cid:82) d x ¯ ψγ ψ .Additionally, the conjugate field momentum correspond-ing to the spinor field ψ ( x ) can be expressed as π ( x ) = ∂ L ψ ∂ ( ∂ ψ ) = − i ¯ ψγ = − iψ † . Therefore, the partition func-tion which contains contributions only from free spinorfield can be expressed using path integral method [22] as Z ψ = (cid:90) D ¯ ψ D ψ e − S βψ , (11)where S βψ = (cid:90) β dτ (cid:90) d x (cid:2) L Eψ + µ ¯ ψ ( τ, x ) γ ψ ( τ, x ) (cid:3) . (12)The Euclidean Lagrangian density L Eψ is obtained by sub-stituting t → iτ in Lagrangian density L ψ and can beexpressed as L Eψ = − ¯ ψ ( τ, x )[ γ ∂ τ + iγ k ∂ k + m ] ψ ( τ, x ) , (13)where k = 1 , ,
3. In the functional integral (11), thespinor field is subject to anti-periodic boundary condi-tions given by ψ ( τ, x ) = − ψ ( τ + β, x ) ; ¯ ψ ( τ, x ) = − ¯ ψ ( τ + β, x ) . (14) It is convenient to express the partition function using theMatsubara frequencies and wave-vector by transformingthe field in Fourier domain as ψ ( τ, x ) = 1 √ V (cid:88) n, k e i ( ω n τ +k · x ) ˜ ψ ( n, k) , (15)where V denotes the spatial volume of the box. The Eq.(14) then implies that the Matsubara frequencies are ω n = (2 n + 1) π β − , (16)where n is an integer. The Eqs. (12) and (15) then leadto S βψ = (cid:88) n, k ¯˜ ψ β (cid:2) /p − m (cid:3) ˜ ψ , (17)where p µ = ( p , (cid:126)p ) = ( − iω n + µ, k) and /p = γ µ p µ . Thespinor fields ψ and ¯ ψ satisfy the same algebra as theGrassmann variables. Using Dirac representation of thegamma matrices and the result of Gaussian integral overGrassmann variables we getln Z ψ = βV π (cid:20) µk F − m ¯ k F + 48 µk F β (cid:21) , (18)where ¯ k F ≡ µk F − m ln (cid:18) µ + k F m (cid:19) . (19)In the Eq. (18), we have also ignored higher order tem-perature corrections which are at least O (( βµ ) − ). B. Partition function for photons
Due to the gauge symmetry A µ ( x ) and A (cid:48) µ ( x ) = A µ ( x ) − e ∂ µ α ( x ) represent the same physical configu-ration. Therefore, in order to avoid over-counting inevaluating the partition function using functional inte-gral methods, it is convenient to introduce the Faddeev-Popov ghost fields C and ¯ C [23, 24]. These Grassmann-valued fields effectively cancel the contributions from twogauge degrees of freedom. Therefore, the thermal parti-tion function containing contributions from the physicalphotons can be expressed as Z A = (cid:90) (cid:16) D A µ e − S βA (cid:17) (cid:16) D ¯ C D C e − S βC (cid:17) ≡ Z A (cid:48) Z C , (20)where S βA = (cid:82) β dτ (cid:82) d x [ − F µν F µν − ξ ( ∂ µ A µ ) ] | t = iτ with gauge-fixing parameter ξ and S βC = (cid:82) β dτ (cid:82) d x [ ∂ µ ¯ C∂ µ C ] | t = iτ . Unlike the spinor field,both A µ ( x ) and C ( x ) fields are subject to the periodicboundary conditions A µ ( τ, x ) = A µ ( τ + β, x ) ; C ( τ, x ) = C ( τ + β, x ) . (21)As earlier, we evaluate the partition function in Fourierdomain by transforming the field as A µ ( τ, x ) = (cid:114) βV (cid:88) n, k e i ( ω n τ +k · x ) ˜ A µ ( n, k) . (22)The definition (22) ensures that the Fourier modes˜ A µ ( n, k) are dimensionless and the Eq. (21) implies ω n = 2 nπβ − with n being an integer. By choosing Feyn-man gauge ξ = 1 and dropping the boundary terms, wecan express S βA as S βA = (cid:88) n, k ¯˜ A µ (cid:20) − β ( ω n + k ) (cid:21) ˜ A µ . (23)Using the identity for Riemann integrals (cid:82) x . . . x N e − x i D ij x j = π N/ ( det ( D )) − / , we canevaluate the contributions from the gauge field byln Z A (cid:48) = − (cid:88) n, k ln (cid:2) β ( ω n + k ) } (cid:3) (24)where the gauge field is Wick rotated as A k → iA k for k = 1 , , C ( τ, x ) = (cid:114) βV (cid:88) n, k e i ( ω n τ +k · x ) ˜ C ( n, k) , (25)where the modes ˜ C ( n, k) are again dimensionless and ω n = 2 nπβ − with n being an integer. By droppingthe boundary terms, we can the express S βC as S βC = (cid:88) n, k ¯˜ C (cid:2) β ( ω n + k ) (cid:3) ˜ C . (26)The ghost fields C and ¯ C being Grassmann-valued field,we can use the same identity as used for fermions in orderto evaluate their contributions asln Z C = (cid:88) n, k ln (cid:2) β ( ω n + k ) } (cid:3) . (27)By combining the contributions (24) and (27) one canwrite the partition function for the physical photons asln Z A = V π β . (28) C. Contributions from the interactions
Including both kinds of interaction for the degenerateelectrons, we can express total partition function as Z = (cid:90) D ¯ ψ D ψ D A µ D ¯ C D C e − ( S βψ + S βA + S βC + S βI ) , (29) where S βI = S β − + S β + with S β − = (cid:82) β dτ (cid:82) d x [ L − I ] | t = iτ and S β + = (cid:82) β dτ (cid:82) d x [ L + I ] | t = iτ . Using perturbativemethod, the total partition function (29) can be ex-pressed as ln Z = ln Z ψ + ln Z A + ln Z I , (30)where the contribution due to the interactions isln Z I = ln (cid:32) ∞ (cid:88) l =1 l ! (cid:104) ( − S βI ) l (cid:105) (cid:33) . (31)Including only the leading order terms we can expressthe Eq. (31) asln Z I = 12 (cid:104) ( S β − ) (cid:105) − (cid:104) ( S β + ) (cid:105) + O ( e ) , (32)where the symbol (cid:104) . (cid:105) denotes the ensemble average.
1. Finite-temperature propagators
In order to compute ln Z I , one needs the finite-temperature propagators for the spinor field and theMaxwell’s field. In particular, the finite-temperaturepropagator for spinor field in real space is defined as G (∆ τ, ∆ x ) = (cid:104) ψ ( τ , x ) ψ ( τ , x ) (cid:105) , (33)where ∆ τ = τ − τ , ∆ x = x − x . The correspondingpropagator in Fourier space is defined as G ( ω n , k) = (cid:90) β dτ (cid:90) d x e − i ( ω n τ +k · x ) G ( τ, x ) . (34)Using the Eq. (17), the propagator for the free spinorfield in Fourier space can be obtained as G ( ω n , k) = 1 /p − m = − /p + mp + m , (35)where p µ = ( p , (cid:126)p ) = ( − iω n + µ, k), /p = γ µ p µ and p = η µν p µ p ν . Similarly, the finite-temperature prop-agator for Maxwell’s field in real space is defined as D µν (∆ τ, ∆ x ) = (cid:104) A µ ( τ , x ) A ν ( τ , x ) (cid:105) . Following theEq. (34), the propagator for free Maxwell fields in Fourierspace can be obtained using the Eq. (23) as D µν ( ω n , k) = − η µν ω n + k . (36)We may emphasize here that in the Eq. (35), the Mat-subara frequencies are ω n = (2 n + 1) πβ − whereas in theEq. (36), they are ω n = 2 nπβ − with n being an integer.
2. Electron-electron interaction
Using the Eq. (7), we can express leading order con-tributions due to the self-interaction of the electrons as (cid:104) ( S β − ) (cid:105) = − e (cid:90) β dτ dτ (cid:90) d x d x D µν (∆ τ, ∆ x ) × Tr (cid:2) γ µ G (∆ τ, ∆ x ) γ ν G ( − ∆ τ, − ∆ x ) (cid:3) , (37)where the trace is carried over the Dirac indices. Herewe have dropped the divergent diagrams that arise fromthe usage of the Wick’s theorem. The Eq. (37), can beexpressed in terms of the propagators in Fourier space as (cid:104) ( S β − ) (cid:105) = − e βV (cid:88) n ,n , k , k D µν (∆ ω n , ∆k) × Tr [ γ µ G ( ω n , k ) γ ν G ( ω n , k )] , (38)where ∆ ω n = ω n − ω n and ∆k = k − k . Using theEqs. (35, 36) one can simplify the Eq. (38) as (cid:104) ( S β − ) (cid:105) = − e βV (cid:88) n ,n k , k m + 2 p · p ( p + m )( p − p ) ( p + m ) , (39)where p = ( − iω n + µ, k ) and p = ( − iω n + µ, k ).Here we have used the trace identities for the Dirac ma-trices Tr( γ µ γ ν ) = − η µν , Tr( γ µ γ ν γ ρ γ σ ) = 4( η µν η ρσ − η µρ η νσ + η µσ η νρ ) and the fact that ( p − p ) = i ∆ ω n .The Eq. (39) can be written in four parts as (cid:104) ( S β − ) (cid:105) = − e βV ( S + S + S + S ) , (40)where S = (cid:88) n ,n , k , k p + m )( p − p ) , (41) S = (cid:88) n ,n , k , k p + m )( p − p ) , (42) S = (cid:88) n ,n , k , k − p + m )( p + m ) , (43) S = (cid:88) n ,n , k , k m ( p + m )( p − p ) ( p + m ) . (44)It can be shown that the term S is infrared divergentand hence ignored. Further, using the symmetries of theexpressions, we may note that S = S = I I , S = − ( I ) where I = (cid:88) n, k p + m , I = (cid:88) n, k p − p ) . (45)Despite the appearance of p in its expression, the eval-uated I does not depend on p and is given by I = (cid:88) n, k β / n π + ( β k2 ) = (cid:88) k β | k | ( + e β | k | − ) = V β . (46) In order to carry out the summation over Matsubara fre-quencies, we have used the identitycoth z = ∞ (cid:88) n = −∞ zn π + z . (47)The summation over k is carried out by converting it toan integral as earlier. Subsequently, by using the Rie-mann zeta function identity ζ (2) = (cid:82) ∞ dt t/ ( e t −
1) = π / I . In order to evaluate I we can ex-press it as I = (cid:88) n, k ω (cid:20) p + ω + 1 − p + ω (cid:21) ≡ I +0 + I − , (48)where p = − i (2 n + 1) π β − + µ and I ± = (cid:88) n, k β ω (cid:18) z ± n π + ( z ± ) (cid:19) , (49)with z ± = { β ( ω ± µ ) ∓ iπ } . By using the identity (47),the summation over n can be carried out as I ± = (cid:88) k β ω (cid:20) − e β ( ω ± µ ) + 1 (cid:21) . (50)The anti-particle contributions are contained in the term I +0 . So by ignoring the anti-particle contributions, thedivergent zero-point energy and by using the approxima-tion βµ (cid:29) I can be evaluated as I = − βV ¯ k F / π .Therefore, the ensemble average becomes (cid:104) ( S β − ) (cid:105) = βV e ¯ k F π (cid:18) ¯ k F π + 13 β (cid:19) . (51)We note that the Eq. (51) differs from an analogous ex-pression, describing the contributions from the electron-electron interaction, given in the textbook by Kapustaand Gale (Eq. 5.59) [22]. However, the expression inthe textbook is erroneous as it implies that electromag-netic repulsion between the electrons causes a reductionof pressure for a system of degenerate electrons in theultra-relativistic regime. In particular, if one ignorestemperature corrections, in the ultra-relativistic limit( k F (cid:29) m ) rhs of the Eq. (51) varies as k F whereas thetextbook expression varies as − k F . On the other hand,in non-relativistic limit ( k F (cid:28) m ), the Eq. (51) variesas k F /m whereas the textbook expression varies as k F .We note that the textbook expression which describespressure corrections due to repulsive electron-electron in-teraction, changes sign as one goes from relativistic tonon-relativistic regime. This aspect itself signals internalinconsistency of the expression given in the textbook.
3. Electron-nuclei interaction
The leading order contribution due to the electron-nuclei interaction can be expressed as (cid:104) S β + (cid:105) = − Ze d (cid:90) β dτ (cid:90) d x J µ ( τ, x ) (cid:104) ψ ( τ, x ) γ µ ψ ( τ, x ) (cid:105) . (52)In order to evaluate the integral (52), it is convenient toexpress it in the Fourier domain as (cid:104) S β + (cid:105) = − Ze d ˜ J µ ( β ) (cid:88) n, k Tr [ γ µ G ( ω n , k)] , (53)where the average background 4-current density˜ J µ ( β ) = 1 βV (cid:90) β dτ (cid:90) d x J µ ( τ, x ) . (54)Within the given box, the spatial motion of the heav-ier nuclei can be neglected. So we may assume thatthe average background 3-current density ˜ J k ( β ) = 0 for k = 1 , ,
3. By identifying the average background chargedensity n + = ˜ J ( β ) = − ˜ J ( β ) and by using the traceidentity of Dirac matrices, we can express the Eq. (53)as (cid:104) S β + (cid:105) = − Ze d n + I , I = (cid:88) n, k p p + m . (55)Similar to the Eq. (48), I can be expressed as I = (cid:88) n, k (cid:20) − p + ω − p + ω (cid:21) ≡ I − − I +2 , (56)and the summation over the Matsubara frequencies canbe carried out as I ± = (cid:88) k β (cid:20) − e β ( ω ± µ ) + 1 (cid:21) . (57)As earlier, by ignoring the anti-particle contributions I +2 ,the divergent zero-point energy and by using the approx-imation βµ (cid:29)
1, finite part of I can be expressed as I = − βV k F / π . The ensemble average (cid:104) S β + (cid:105) thenbecomes (cid:104) S β + (cid:105) = βV Ze d k F n + π . (58)
4. Total contributions from the interactions
The number density of positively charged nuclei mustsatisfy Zn + = n e as system is overall electrically neutral.Therefore, by combining the contributions from the self-interaction of the electrons (51) and the electron-nucleiinteraction (58), we can express the partition function(32) due to the total interaction asln Z I = βV e π (cid:0) k F − π d n e k F (cid:1) , (59) where we have ignored the finite temperature correctionswithin the parenthesis as the coupling constant e and theterm ( βk F ) − both are small. IV. EQUATION OF STATE
Using the evaluated partition function we can computethe pressure and the mass density within the consideredbox located at the given radial coordinate. Subsequently,we may read off the corresponding equation of state ofthe degenerate matter at the given radial location. Forlater convenience, we now define following dimensionlessparameters σ ≡ mk F , σ µ ≡ µk F , σ k ≡ ¯ k F k F , σ T ≡ k B Tk F . (60)We note that σ − , as defined here, can be identified withthe so called ‘relativity parameter’ in the literature [6].We also note that σ µ = √ σ , σ k = σ µ − σ log((1 + σ µ ) /σ ) and σ T = ( βk F ) − . For typical white dwarfs, theEq. (1) implies σ T (cid:28)
1. For a system of ultra-relativisticdegenerate electrons σ (cid:28) σ µ (cid:39) σ k (cid:39) A. Mass density
The number density of the electrons can be com-puted from total partition function as n e ≡ (cid:104) N (cid:105) /V =( βV ) − ( ∂ ln Z /∂µ ). Given the partition function due tothe interaction terms (59) itself depends on the electronnumber density, it leads to an algebraic equation for n e as given below n e = k F π (cid:20) σ (6 σ T ) − + 3 α π (cid:26) σ k − π d n e σ µ k F (cid:27)(cid:21) . (61)In order to arrive at the Eq. (61), we have used two veryuseful relations ( ∂k F /∂µ ) = ( µ/k F ) and ( ∂ ¯ k F /∂µ ) =2 k F . The Eq. (61) can be solved in a straightforwardmanner to result n e = k F π (cid:34) σ T (2 + σ ) + (3 α/ π ) σ k d ( α/π ) σ − µ (cid:35) , fine structure constant is α = e / π in nat-ural units. By using the chemical potential µ whichcomes naturally in the partition function (10), we havedefined a dimensionless parameter d + ≡ dµ . The pa-rameter d + characterizes the associated length scale withthe electron-nuclei interaction and it needs to be fixed byseparate consideration (see FIG. 1). Now the mass den-sity of the system is given by ρ = µ e m u n e , (63)where m u is the atomic mass unit. The parameter µ e ≡ ( A/Z ) is defined so that µ e m u specifies ‘the average mass N u m be r den s i t y ( i n k F / π ) m/k F d = 0.1 µ -1 d = 0.5 µ -1 d = 1.5 µ -1 FIG. 1: The number density of electrons at zero temperaturefor different values of length scale d . per electron’ where A is the atomic mass number. For awhite dwarf with pure Helium He core µ e is 2. B. Pressure
Using the expression of pressure for a grand canoni-cal ensemble, we may read off the pressure due to thedegenerate electrons as P ψ = ( βV ) − ln Z ψ and due tothe interactions as P I = ( βV ) − ln Z I . One may checkthat the radiation pressure P A = π β − is insignifi-cant even compared to P I , as for white dwarfs βk F (cid:29) P = k F π (cid:20) σ T σ − µ − σ σ k α π ( σ k − π d n e k F σ µ ) (cid:21) . (64)We may again note that the degeneracy pressure dependson the parameter d + which characterizes the electron-nuclei interaction length scale (see FIG. 2). Using theEqs. (62, 63, 64), in principle, one can express theequation of state for the degenerate matter within whitedwarfs as P = P ( ρ ) which includes the corrections due tothe fine structure constant α and the finite temperature. C. Non-relativistic limit
We have the considered matter field actions to be man-ifestly Lorentz invariant here. Consequently the studiedequation of state is well suited for describing the relativis-tic regime. However, for the consistency, the equation ofstate must also have correct non-relativistic limit when k F (cid:28) m . In such limit σ (cid:29) σ µ = σ + σ − − σ − + O ( σ − ) and σ k = σ − − σ − + O ( σ − ). Therefore,in the non-relativistic regime, the number density (62) P r e ss u r e ( i n k F / π ) m/k F d = 0.1 µ -1 d = 0.5 µ -1 d = 1.5 µ -1 FIG. 2: The degeneracy pressure at zero temperature for dif-ferent values of length scale d . reduces to n e (cid:39) k F π (cid:20) m k B T k F + α (1 − d ) k F πm (cid:21) , (65)and the pressure (64) reduces to P (cid:39) k F π m (cid:20) m k B T k F + 2 α (1 − d ) k F πm (cid:21) . (66)If one disregards the corrections due to the finite tem-perature and the fine-structure constant, the Eqs. (65,66) represent the standard non-relativistic expressions.However, one may note that in the non-relativistic regimethe effects of finite temperature become important. Nev-ertheless, these equations are valid in non-relativisticregime as long as corresponding chemical potential µ sat-isfies βµ (cid:29) D. Temperature corrections
For non-interacting, zero temperature degenerate elec-tron gas, the number density of electrons is given by n e = ( k F / π ). However, the effect of finite temper-ature causes this relation to be modified even for non-interacting electrons as n e ( T ) n e ( T = 0) = 1 + 6(2 + σ ) k B T k F . (67)Analogously, the effect of finite temperature on the pres-sure of non-interacting degenerate electron gas can beexpressed as P ( T ) P ( T = 0) = 1 + 48 σ µ k B T k F (2 σ µ − σ σ k ) . (68)Clearly, the finite temperature causes the pressure to in-crease for a given Fermi momentum k F . However, theincrease in pressure is very small given it is of the order ∼ ( βk F ) − ∼ − for typical white dwarfs (1). E. Fine-structure constant corrections
The effects of the electromagnetic interaction i.e. theCoulomb effects on the equation of state are expressedusing the fine-structure constant α (cid:39) /
137 which isa small number. However, theses corrections are muchlarger compared to the temperature corrections. At thezero temperature, the leading order effect of the fine-structure constant on the electron number density canbe expressed as n e ( α ) n e ( α = 0) = 1 + α π (cid:0) σ k − d σ − µ (cid:1) . (69)Similarly, at the zero temperature the leading order ef-fect of the fine-structure constant on the pressure can beexpressed as P ( α ) P ( α = 0) = 1 + α π (9 σ k − d σ − µ )(2 σ µ − σ σ k ) . (70)We note that the number density and the pressure bothcontain an undetermined dimensionless parameter d + =2 dµ in the corrections involving the fine-structure con-stant. As mentioned earlier, the length scale d is asso-ciated with the current-current interaction between theelectrons and the nuclei. In the partition function, a nat-ural length scale is provided by the chemical potential µ .Therefore, intuitively one would expect that the dimen-sionless parameter d + to be an O (1) number for the whitedwarfs. However, determination of its exact numericalvalues can only be done by using separate considerations,possibly by using observations. In the standard litera-ture, this one-parameter uncertainty is often overlookedas usually there one fixes the lattice scale associated withpositively charged nuclei by heuristic arguments. How-ever, we have argued that this length scale is associatedwith the electron-nuclei interaction and its independentdetermination in principle can allow one to understandthe property of the underlying lattice structure formedby the nuclei within the degenerate matter of the whitedwarfs. F. Comparison with Salpeter’s corrections
In order to compare the number density (62) and thepressure (64) with that of Salpeter’s we need to set σ T = 0 as these are studied at zero temperature bySalpeter [6]. Further, for comparison we consider theterms up to leading order in fine structure constant α from the combined expressions of non-interacting de-generacy pressure P , classical Coulomb corrections P C ,Thomas-Fermi corrections P T F , exchange corrections P ex and correlation corrections P cor as described in [6].In the treatment by Salpeter, the relation between thenumber density of electrons n e and the Fermi momentum k F is assumed to be fixed. On the other hand, the usageof grand canonical partition function here implies that P r e ss u r e ( i n k F / π ) m/k F No interaction: α =0Relativistic QFT (d=1.18 µ -1 )Salpeter (Z=6) FIG. 3: A comparison of the pressure in a broadly relativisticdomain. there is a modification to the expression of the electronnumber density due to the electromagnetic interactions.In turns, it would imply a difference in equation of stateeven if the pressure expressions considered by Salpeterand here, were to agree.
1. No interaction
The expressions of the number density (62) and thepressure (64) agree exactly with the Salpeter’s expres-sions when one ignores the fine-structure constant cor-rections by setting α = 0 and identifies σ − as the ‘rela-tivity parameter’ x along with the mathematical identitysinh − x = ln( x + √ x ).
2. Relativistic domain
In the ultra-relativistic limit, k F (cid:29) m , we can expressthe total pressure which includes leading order correc-tions due to the fine-structure constant, as P = k F π (cid:20) α (cid:18) π − d π (cid:19)(cid:21) . (71)On the other hand, analogous expression for pressurewith leading order corrections considered by Salpeter canbe expressed as P = k F π (cid:2) α ( π − ( π ) / Z / ) (cid:3) [6]. Therefore, if one chooses d = + ( π ) / Z / then one would get the same pressure corrections in theultra-relativistic limit. In particular, if one chooses theatomic number Z = 2 (Helium) or Z = 6 (Carbon)then the Salpeter’s corrections would correspond to thelength scale d being 0 . µ − , 1 . µ − respectively. Thisis in agreement with the intuitive expectation that d + should be an O (1) number. The pressure comparison ina broadly relativistic domain is given in the FIG. 3.Nevertheless, we emphasize that the length scale d isundetermined apriori in the approach that we have con-sidered here. For a given system of degenerate electrons -0.02 0 0.02 0.04 0.06 0.08 0.1 10 100 P r e ss u r e ( i n k F / π ) m/k F No interaction: α =0Relativistic QFT (d=1.18 µ -1 )Salpeter (Z=6) FIG. 4: A comparison of the pressure in a broadly non-relativistic domain. The pressure becoming negative signalsthe breakdown of the underlying assumptions in Salpeter’streatment. and nuclei, in principle, it may be possible to derive an effective action corresponding to Eq. (8) where ∼ d + e may be viewed as the renormalized coupling constant be-tween the electrons and the nuclei at the energy scale setby the chemical potential µ .
3. Non-relativistic domain
The corrections to the pressure expression, consideredby Salpeter, are known to become unreliable in a fairlynon-relativistic domain. Salpeter noted that with suchcorrections the total pressure could become negative, sig-naling the breakdown of the underlying assumptions [6].In contrast, the non-relativistic expression (66) here re-mains well defined. A comparison of the pressure in abroadly non-relativistic domain is given in the FIG. 4.
G. Modified mass limit of white dwarfs
In order to understand the effect of the fine-structureconstant on the Chandrasekhar mass limit for whitedwarfs, we need to find the corrections to the equation ofstate in the ultra-relativistic limit. In such limit, the Eqs.(62, 63, 64) together lead to a polytropic equation of stateof the form P ∝ ρ / where fine-structure constant mod-ifies the proportionality constant. Such a modificationin turn leads to the modified Chandrasekhar mass limit M ch which including up to the leading order correctionin α , is given by [25] M ch M ch = 1 − α π , (72)where M ch is the Chandrasekhar mass limit without fine-structure constant corrections. We note that the effect ofthe fine-structure constant α reduces the Chandrasekharmass limit for white dwarfs by a universal factor [25].In particular, the length scale d which is associated with the electron-nuclei interaction, does not affect the masslimit for the white dwarfs. In contrast, the modifiedChandrasekhar mass limit which uses the Salpeter’s cor-rections, can be expressed up to leading order in α , as M ch /M ch = 1 − α (cid:2) ( π ) / Z / − π (cid:3) [6–8]. So thereduction of the Chandrasekhar mass limit there is non-universal as it depends on the atomic number of the con-stituent nuclei. V. DISCUSSION
In the literature, the effects of the electromagnetic in-teraction on the equation of state of the degenerate mat-ter within the white dwarfs are computed by consider-ing the so-called classical Coulomb energy, the Thomas-Fermi effect, the exchange and correlation energy at zerotemperature. These computations rely on the electro-static considerations which are non-relativistic ab ini-tio. In this article, after reviewing the existing litera-ture, we have presented a computation of the equationof state of degenerate matter for the white dwarfs byemploying the techniques of finite temperature relativis-tic quantum field theory. The corresponding equation ofstate includes the leading order effect due to the fine-structure constant and the effect of finite temperature.The correction to the equation of state due to the fine-structure constant has two components. The first compo-nent arises from the self-interaction between the degen-erate electrons and described by the action of quantumelectrodynamics. For the second component we have con-sidered a Lorentz invariant interaction term to describethe interaction between electrons and positively chargednuclei. Further, we have argued that a fully relativis-tic consideration leads to an apriori undetermined lengthscale in the corrections to the equation of state involv-ing the electron-nuclei interaction. This aspect of theequation state is overlooked in the literature. Insteadthere one fixes the associated scale by using heuristic ar-guments. An independent determination of this lengthscale may shed light on the underlying lattice structureformed by the nuclei within the degenerate matter of thewhite dwarfs. Besides, the effect of fine-structure con-stant reduces the Chandrasekhar mass limit of the whitedwarfs by a universal factor which is independent of theatomic number of the constituent nuclei and the electron-nuclei interaction length scale.In order to describe the background geometry withinthe white dwarfs, here we have used the Minkowski space-time. In other words, we have ignored the effects of gen-eral relativity as one expects the effect of gravity on theequation of state to be smaller than the effect of the fine-structure constant for the observed white dwarfs. How-ever, it would be interesting in its own right to considerthe effects of general relativity within the considered ap-proach as done for other approaches [13, 26] or even fordifferent applications [27].Finally, we note that the future detection of low-0frequency gravitational waves from the extreme mass-ratio merger of a black hole and a white dwarf coulddetermine the equation of state of the degenerate matterwithin the white dwarf with an accuracy reaching up to0 .
1% [17]. It is based on the expectation that the tidaldisruption of a white dwarf, during the final phase ofinspiral around a massive black hole, could be measuredvery accurately through low-frequency gravitational wavesignals. The properties of the tidal disruption of a whitedwarf would necessarily depend on the equation of stateof the degenerate matter present within the white dwarf,rather than its maximal mass limit. On the other hand,we may note from the Eqs. (69, 70) that the correc-tions to the number density and the pressure due to thefine-structure constant are of the order ∼ α/π ∼ . Acknowledgments
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