Antimatter Gravity: Second Quantization and Lagrangian Formalism
aa r X i v : . [ phy s i c s . g e n - ph ] J u l Antimatter Gravity: Second Quantization andLagrangian Formalism
Ulrich D. Jentschura
Department of Physics, Missouri Universityof Science and Technology, Rolla, Missouri 65409, USA; [email protected]–DE Particle Physics Research Group,P.O. Box 51, H–4001 Debrecen, HungaryMTA Atomki, P.O. Box 51, H–4001 Debrecen, Hungary
July 21, 2020
Abstract
The application of the CPT theorem to an apple falling on Earth leads to thedescription of an anti-apple falling on anti–Earth (not on Earth). On the micro-scopic level, the Dirac equation in curved space-time simultaneously describesspin-1 / Keywords: General Relativity, Antimatter Gravity, Antiparticles, CPT Symmetry,Spin Connection
It is common wisdom in atomic physics that the Dirac equation describes particlesand antiparticles simultaneously, and that the negative-energy solutions of the Diracequation have to be reinterpreted in terms of particles that carry the opposite chargeas compared to particles, and whose numerical value of the energy E is equal to the1egative value of the physically observed energy [1]. Based on the Dirac equation,the existence of the positron was predicted, followed by its experimental detectionin 1933, by Anderson [2]. If we did not reinterpret the negative-energy solutionsof the Dirac equation, then the helium atom would be unstable against decay intoa state where the two electrons perform quantum jumps into continuum states [3].One of the electrons would jump into the positive-energy continuum, the other, intothe negative-energy continuum, with the sum of the energies of the two continuumstates being equal to the sum of the two bound-state energies of the helium atomfrom which the transition started [3, 4, 5].The absolute necessity to reinterpret the negative-energy solutions of the Dirac equa-tion as antiparticle wave functions, i.e., the necessity to interpret positive-energy andnegative-energy solutions of one single equation as describing two distinct particles,hints at the possibility to use the Dirac equation as a bridge to the description ofthe gravitational interaction of antimatter. Namely, if the Dirac equation is beingcoupled to a gravitational field, then, since it describes particles and antiparticlessimultaneously, the Dirac equation offers us an additional dividend: In addition todescribing the gravitational interaction of particles, the Dirac equation automat-ically couples the antiparticle (the “anti-apple”), which is described by the sameequation, to the gravitational field, too.Corresponding investigations have been initiated in a series of recent publications [6,7, 8, 9]. One may ask whether the dynamics of particles and antiparticles differ ina central, static, gravitational field, in first approximation, but also, if there areany small higher-order effects breaking the particle-antiparticle symmetry underthe gravitational interaction. The first of these questions has been answered inRefs. [6, 7, 8], with the result being that the Dirac particle and antiparticle behaveexactly the same in a central gravitational field, due to a perfect particle-antiparticlesymmetry which extends to the relativistic and curved-space-time corrections to theequations of motion.In order to address the second question, it is necessary to perform the full particle-to-antiparticle symmetry transformation of the Dirac formalism, in an arbitrary (pos-sibly dynamic) curved-space-time-background. This transformation is most strin-gently carried out on the level of the Lagrangian formalism. A preliminary resulthas recently been published in Ref. [9], where a relationship was established be-tween the positive-energy and negative-energy solutions of the Dirac equation inan arbitrary dynamics curved-space-time-background. However, the derivation inRef. [9] is based on a first-quantized formalism, which lacks the unified descriptionin terms of the field operator. The field operator comprises all (as opposed to any )solution of the gravitationally (and electromagnetically) coupled Dirac equation. Ingeneral, a satisfactory description of antiparticles, in the field-theoretical context,necessitates a description in terms of particle- and antiparticle creation and annihi-lation processes, and therefore, the introduction of a field operator. In consequence,the investigation [9] is augmented here on the basis of a transformation of the en-tire Lagrangian density, which can be expressed in terms of the charge-conjugated(antiparticle) bispinor wave function, and generalized to the level of second quanti-zation. The origin [9] of a rather disturbing minus sign which otherwise appears in2he Lagrangian formalism upon charge conjugation in first quantization will be ad-dressed. The use of the Lagrangian formalism necessitates a definition of the Diracadjoint in curved space-times. As a spin-off result of the augmented investigationsreported here, we find the general form of the Dirac adjoint in curved space-times,in the Dirac representation of γ matrices.According to Ref. [10], the role of the CPT transformation in gravity needs to be con-sidered with care: A priori , a CPT transformation of a physical system consisting ofan apple falling on Earth would describe the fall of an anti-apple on anti-Earth. Keyto our considerations is the fact that, on the microscopic level, the Dirac equationapplies (for one and the same space-time metric) to both particles and antiparti-cles simultaneously (this translates, on the macroscopic level, to “apples” as wellas “anti-apples”). This paper is organized as follows: We investigate the generalform of the Dirac adjoint in Sec. 2, present our theorem in Sec. 3, and in Sec. 4,we provide an overview of connections to new forces and CPT violating parameters,Conclusions are reserved for Sec. 5.
In order to properly write down the Lagrangian of a Dirac particle in a gravitationalfield, we first need to generalize the concept of the Dirac adjoint to curved space-times. We recall that the Dirac adjoint transforms with the inverse of the Lorentztransform as compared to the original Dirac spinor. A general spinor Lorentz trans-formation S (Λ) is given as follows, S (Λ) = exp (cid:18) − i4 ǫ AB σ AB (cid:19) , σ AB = i2 (cid:2) γ A , γ B (cid:3) , A, B = 0 , , , . (1)Note that the generator parameters ǫ AB = − ǫ BA , for local Lorentz transformations,can be coordinate-dependent. In the following, capital Roman letters A, B, C, · · · =0 , , , γ A are assumed to be taken in the Dirac representation [1], γ = (cid:18) × × (cid:19) , ~γ = (cid:18) ~σ − ~σ (cid:19) . (2)Here, the vector of Pauli spin matrices is denoted as ~σ . In consequence, the spin ma-trices σ AB are the flat-space spin matrices. The spin matrices fulfill the commutationrelations[ σ CD , σ EF ] = i (cid:0) g CF σ DE + g DE σ CF − g CE σ DF − g DF σ CE (cid:1) . (3)These commutation relations, we should note in passing, are completely analogousto those fulfilled by the matrices M AB that generate (four-)vector local Lorentztransformations. As is well known, the latter have the components (denoted byindices C and D ) ( M AB ) C D = g C A g DB − g C B g DA . (4)3he vector local Lorentz transformation Λ with components Λ C D is obtained as thematrix exponential Λ
C D = (cid:18) exp (cid:20) ǫ AB M AB (cid:21)(cid:19) C D . (5)The algebra fulfilled by the M matrices is well known to be[ M CD , M EF ] = g CF M DE + g DE M CF − g CE M DF − g DF M CE . (6)The two algebraic relations (3) and (6) are equivalent if one replaces M CD → − i2 σ CD , (7)which leads from Eq. (1) to Eq. (5). Under a local Lorentz transformation, a Diracspinor transforms as ψ ′ ( x ′ ) = S (Λ) ψ ( x ) . (8)In order to write the Lagrangian, one needs to define the Dirac adjoint in curvedspace-time. In order to address this question, one has to remember that in flat-space-time, the Dirac adjoint ψ ( x ) is defined in such a way that is transforms withthe inverse of the spinor Lorentz transform as compared to ψ ( x ), ψ ′ ( x ′ ) = ψ ( x ) S (Λ − ) = ψ ( x ) [ S (Λ)] − . (9)The problem of the definition of ψ ( x ) in curved space-time is sometimes treated inthe literature in a rather cursory fashion [11]. Let us see if in curved space-time, wecan use the ansatz ψ ( x ) = ψ + ( x ) γ , (10)with the same flat-space γ as is used in the flat-space Dirac adjoint. In this case, ψ ′ ( x ′ ) = ψ + ( x ′ ) S + (Λ) γ = (cid:16) ψ + ( x ′ ) γ (cid:17) h γ S + (Λ) γ i , (11)To first order in the Lorentz generators ǫ AB , we have indeed, γ S + (Λ) γ = 1 + i4 ǫ AB γ σ + AB γ = 1 + i4 ǫ AB σ AB = [ S (Λ)] − , (12)where we have used the identity σ + AB = − i2 [ γ + B , γ + A ] = − i2 γ [ γ γ + B γ , γ γ + A γ ] γ = − i2 γ [ γ B , γ A ] γ = − γ σ BA γ = γ σ AB γ . (13)It is easy to show that Eq. (12) generalizes to all orders in the ǫ AB parameters, whichjustifies our ansatz given in Eq. (10). The result is that the flat-space γ matrixcan be used in curved space, just like in flat space, in order to construct the Diracadjoint. The Dirac adjoint spinor transforms with the inverse spinor representationof the Lorentz group [see Eq. (9)]. 4 Lagrangian and Charge Conjugation
Equipped with an appropriate form of the Dirac adjoint in curved space-time, westart from the Lagrangian density [12, 13, 14, 11, 15, 16, 17, 18, 19, 20] L = ψ ( x ) [ γ µ { i ( ∂ µ − Γ µ ) − e A µ } − m I ] ψ ( x ) , (14)Here, the A µ field describes the four-vector potential of the electromagnetic field,while the Γ µ matrices describe the spin connection.Γ µ = i4 ω ABµ σ AB , ω ABµ = e Aν ∇ µ e νB , ∇ µ e νB = ∂ µ e νB + Γ νµρ e ρB . (15)For the form of the covariant coupling, we refer to Eqs. (3.129) and (3.190) ofRef. [20]. In the above equations, capital Roman indices A, B, C, · · · = 0 , , , µ, ν, ρ, · · · = 0 , , , µ matriceswhich describe the gravitational coupling. In other words, we note that none of thementioned standard textbooks of quantum field theory discuss the gravitationally coupled Dirac equation, and all cited descriptions are limited to the flat-space Diracequation, where the role of the charge conjugation operation is much easier to analyzethan in curved space.The double-covariant coupling to both the gravitational as well as the electromag-netic field is given as follows, D µ = ∂ µ − Γ µ + i e A µ = ∇ µ + i e A µ , (16)where ∇ µ = ∂ µ − Γ µ is the gravitational covariant derivative.As a side remark, we note that gravitational spin connections Γ µ = i4 ω ABµ σ AB and other gauge-covariant couplings are unified in the so-called spin-charge familytheory [30, 31, 32, 33, 34] which calls for a unification of all known interactionsof nature in terms of an SO (1 ,
13) overarching symmetry group. (In the currentarticle, we use the spin connection matrices purely in the gravitational context.)The SO (1 ,
13) has a 25-dimensional Lie group, with 13 boosts and 12 rotations inthe internal space. This provides for enough Lie algebra elements to describe theStandard Model interactions, and predict a fourth generation of particles. The spin-charge family theory is a significant generalization of Kaluza-Klein-type ideas [35,36].In the context of the current investigations, though, we restrict ourselves to thegravitational spin connection matrices. In view of the (in general) nonvanishing5pace-time dependence of the Ricci rotation coefficients, we can describe the quan-tum dynamics of relativistic spin-1 / σ AB matrices defined in Eq. (15) represent the six generators of the spin-1 / L = L + = ψ + ( x ) h ( γ µ ) + n − i ←− ∂ µ − e A µ o − ( − i) (Γ µ ) + ( γ µ ) + − m I i (cid:2) ψ ( x ) (cid:3) + . (17)An insertion of γ matrices under use of the identity ( γ ) = 1 leads to the relation L + = ψ + ( x ) γ h γ ( γ µ ) + γ n − i ←− ∂ µ − e A µ o +i n γ (Γ µ ) + γ o γ ( γ µ ) + γ − m I i γ (cid:2) ψ ( x ) (cid:3) + . (18)Also, we recall that γ (Γ µ ) + γ = − Γ µ , becauseΓ + µ = − i4 ω ABµ σ + AB = − i4 ω ABµ γ σ AB γ = − γ Γ µ γ . (19)So, the adjoint of the Lagrangian is L + = ψ + ( x ) γ h γ µ n − i ←− ∂ µ − e A µ o − i Γ µ γ µ − m I i γ (cid:2) ψ ( x ) (cid:3) + . (20)Now, we use the relations ψ + ( x ) γ = ψ ( x ) and γ (cid:2) ψ ( x ) (cid:3) + = ψ ( x ), and arrive atthe form L + = ψ ( x ) h γ µ n − i ←− ∂ µ − e A µ o − i Γ µ γ µ − m I i ψ ( x ) . (21)Because L is a scalar, a transposition again does not change the Lagrangian, andwe have (cid:16) L + (cid:17) T = ψ T ( x ) h ( γ µ ) T n − i −→ ∂ µ − e A µ o − i ( γ µ ) T (Γ µ ) T − m I i (cid:2) ψ ( x ) (cid:3) T . (22)An insertion of the charge conjugation matrix C = i γ γ (with the flat-space γ and γ ) leads to (cid:16) L + (cid:17) T = ψ T ( x ) C − h C ( γ µ ) T C − n − i −→ ∂ µ − e A µ o − i C ( γ µ ) T C − C Γ T µ C − − m I i C (cid:2) ψ ( x ) (cid:3) T . (23)we use the identities C ( γ µ ) T C − = − γ µ , and C (Γ µ ) T C − = − Γ µ . The latter ofthese can be shown as follows, C Γ T µ C − = i4 (cid:26) i2 ω ABµ C (cid:2) γ T B , γ T A (cid:3) C − (cid:27) = i4 (cid:26) i2 ω ABµ [ − γ B , − γ A ] (cid:27) = − Γ µ . (24)The result is the expression (cid:16) L + (cid:17) T = ψ T ( x ) C − h ( − γ µ ) n − i −→ ∂ µ − e A µ o − i ( − γ µ ) ( − Γ µ ) − m I i C (cid:2) ψ ( x ) (cid:3) T . (25)6ow we express the result in terms of the charge-conjugate spinor ψ C ( x ) and itsadjoint ψ C ( x ) (further remarks on this point are presented in Appendix B), ψ C ( x ) = C (cid:2) ψ ( x ) (cid:3) T , ψ C ( x ) = − ψ T ( x ) C − , (26)where we use the identity C − = − C (see also Appendix A). The Lagrangian be-comes L = (cid:16) L + (cid:17) T = − ψ C ( x ) h γ µ n i −→ ∂ µ + e A µ o − i γ µ Γ µ − m I i ψ C ( x )= − ψ C ( x ) [ γ µ { i( ∂ µ − Γ µ ) + e A µ } − m I ] ψ C ( x ) . (27)The Lagrangian given in Eq. (27) differs from (17) only with respect to the signof electric charge, as is to be expected, and with respect to the replacement of theDirac spinor ψ ( x ) by its charge conjugation ψ C ( x ). The overall minus sign is phys-ically irrelevant as it does not influence the variational equations derived from theLagrangian; besides, it finds a natural explanation in terms of the reinterpretationprinciple, if we interpret ψ ( x ) as a Dirac wave function in first quantization.Namely, there is a connection of the spatial integrals of the mass term, proportionalto J = Z d r ψ ( x ) ψ ( x ) = Z d r ψ ( t, ~r ) ψ ( t, ~r ) = Z d r ψ + ( t, ~r ) γ ψ ( t, ~r ) , (28)and the charge conjugate, J C = Z d r ψ C ( x ) ψ C ( x ) = Z d r (cid:0) ψ C ( t, ~r ) (cid:1) + γ ψ ( t, ~r ) . (29)Both of the above integrals connect to the energy eigenvalue of the Dirac equation inthe limit of time-independent fields (see Appendices A and B). One can show that theenergy eigenvalues of Dirac eigenstates ψ , in the limit of weak potentials and statescomposed of small momentum components, exactly correspond to the integrals J and J C (up to a factor m I ). In turn, the dominant term in the Lagrangian in thislimit is L → − ψ ( x ) m I ψ ( x ) = + ψ C ( x ) m I ψ C ( x ) . (30)Because the integral R d r L equals − J (or + J C ), the sign change becomes evident:it is due to the fact that the states ψ C describe antiparticle wave functions wherethe sign of the energy flips in comparison to particles. The matching of m I to thegravitational mass can be performed in a central, static field [6, 9], and results inthe identification m I = m G , where m G is the gravitational mass. The gravitationalcovariant derivative ∂ µ − Γ µ has retained its form in going from (17) to (27), inagreement with the perfect particle-antiparticle symmetry of the gravitational inter-action. Because the above demonstration is general and holds for arbitrary (possiblydynamic) space-time background Γ, there is no room for a deviation of the gravita-tional interactions of antiparticles (antimatter) to deviate from those of matter. Thishas been demonstrated here on the basis of Lagrangian methods, supplementing arecent preliminary result [9]. 7n order to fully clarify the origin of the minus sign introduced upon charge conju-gation, one consults Chaps. 2 and 3 of Ref. [1] and Chap. 7 of Ref. [29]. Namely,in second quantization, there is an additional minus sign incurred upon the chargeconjugation, which restores the original sign pattern of the Lagrangian. Accordingto Eq. (2.107) and (3.157) of Ref. [1], we can write the expansion of the free Diracfield operator asˆ ψ ( x ) = X s Z d p (2 π ) mE h a s ( ~p ) u s ( ~p ) e − i p · x + e i p · x v s ( ~p ) b + s ( ~p ) i . (31)The field operator is denoted by a hat in order to differentiate it from the Dirac wavefunction. The four-momentum is p µ = ( E, ~p ), where E = p ~p + m is the free Diracenergy, and u s ( ~p ) and v s ( ~p ) are the positive-energy and negative-energy spinors withspin projection s (onto the z axis). Furthermore, the particle annihilation operator a s ( ~p ) and the antiparticle creation operator b + s ( ~p ), and their Hermitian adjoints,fulfill the commutation relations given in Eqs. (3.161) of Ref. [1], n a s ( ~p ) , a + s ′ ( ~p ) o = Em (2 π ) δ (3) ( ~p − ~p ′ ) δ ss ′ , (32a) n b s ( ~p ) , b + s ′ ( ~p ′ ) o = Em (2 π ) δ (3) ( ~p − ~p ′ ) δ ss ′ . (32b)The spinors are normalized according to Eq. (2.43a) of Ref. [1], i.e., they fulfillthe relation u + s ( ~p ) u s ( ~p ) = v + s ( ~p ) v s ( ~p ) = E/m . For the charge conjugation in thesecond-quantized theory, it is essential that an additional minus sign is incurred inview of the anticommutativity of the field operators. Namely, without consideringthe interchange of the field operators, one would have, under charge conjugation, J µ ( x ) = ψ ( x ) γ µ ψ ( x ) = ψ C ( x ) γ µ ψ C ( x ) = J C µ ( x ), i.e., the current would not changeunder charge conjugation which is intuitively inconsistent [see the remark followingEq. (4.618) of Ref. [29]]. However, for the field operator current (from here on, we de-note field operators with a hat), we have ˆ J µ ( x ) = ˆ ψ ( x ) γ µ ˆ ψ ( x ) = − ˆ ψ C ( x ) γ µ ˆ ψ C ( x ) = − ˆ J C µ ( x ), because one has incurred an additional minus sign due to the restorationof the field operators into their canonical order after charge conjugation [see theremark following Eq. (7.309) of Ref. [29]].In our derivation above, when one transforms to a second-quantized Dirac field(but keeps classical background electromagnetic field and a classical non-quantizedcurved-space-time metric), one starts from Eq. (21) as an equivalent, alternativeformulation of Eq. (14). One observes that in going from Eq. (21) to (22), one hasactually changed the order of the field operators in relation to the Dirac spinors.Restoring the original order, much in the spirit of Eq. (7.309) of Ref. [29], one incursan additional minus sign which ensures thatˆ L = ˆ ψ ( x ) [ γ µ { i ( ∂ µ − Γ µ ) − e A µ } − m I ] ˆ ψ ( x )= ˆ ψ C ( x ) [ γ µ { i( ∂ µ − Γ µ ) + e A µ } − m I ] ˆ ψ C ( x ) , (33)exhibiting the effect of charge conjugation in the second-quantized theory—andrestoring the overall sign of the Lagrangian. The theorem (33) shows that parti-cles and antiparticles behave exactly the same in gravitational fields, but it does8ot imply, a priori , that m I = m G . The matching of the inertial mass m I and thegravitational mass m G most easily proceeds in a central, static field (Schwarzschildmetric), as demonstrated in Sec. 3 of Ref. [9].One should, at this stage, remember that experimental evidence, to the extent pos-sible, supports the above derived symmetry relation. The only direct experimentalresult on antimatter and gravity comes, somewhat surprisingly, from the Supernova1987A. Originating from the Large Magellanic Cloud, the originating neutrinos andantineutrinos eventually were detected on Earth. In view of their travel time ofabout 160,000 years, they were bent from a “straight line” by the gravity from ourown galaxy. The gravitational bending changed the time needed to reach Earth byabout 5 months. Yet, both neutrinos and antineutrinos reached Earth within thesame 12 second interval, shows that neutrinos and antineutrinos fall similarly, toa precision of about 1 part in a million [37, 38]. In view of the exceedingly smallrest mass of neutrinos, the influence of the mass term (even a conceivable tachyonicmass term) on the trajectory is negligible [39]. Yet, it is reassuring that experimentalevidence, at this time, is consistent with Eq. (33). In view of the symmetry relations derived in this article for the gravitationally andelectromagnetically coupled Dirac equation, it is certain justified to ask about anadequate interpretation of antimatter gravity experiments. We have shown thatcanonical gravity cannot account for any deviations of gravitational interactions ofmatter versus antimatter. How could tests of antimatter “gravity” be interpretedotherwise? The answer to that question involves clarification of the question which“new” interactions could possibly mimic gravity. The criteria are as follows: (i)
The“new” interaction would need to violate CPT symmetry. (ii)
The “new” interactionwould have to be a long-range interaction, mediated by a massless virtual particle.One example of such an interaction would be induced if hydrogen atoms were to ac-quire, in addition to the electric charges of the constituents (electrons and protons),an additional “charge” η e , where e is the elementary charge, while antihydrogenatoms would acquire a charge − η e , where η is a small parameter. One could con-jecture the existence of a small, CPT-violating “charge” ηe/ − ηe/
2. We will refer to this concept as the “ η force” in the following. Thedifference in the gravitational force (acceleration due to the Earth’s field) felt by ahydrogen versus an antihydrogen atom is F η H − F η H = 2 η h η N p + N n + N e ) i e πǫ R ⊕ . (34)Here, R ⊕ is the radius of the Earth, while N p , N n and N e are the numbers ofprotons, neutrons and electrons in the Earth. The gravitational force on a fallingantihydrogen atom is F G H = G m p M ⊕ R ⊕ . (35)9et us assume that an experiment establishes that | F η H − F η H | < χ F G H , where χ isa measure of the deviation of the acceleration due to gravity+“ η ”-force for anti-hydrogen versus hydrogen. A quick calculation shows that this translates into abound η < . × − √ χ . (36)Antimatter gravity tests thus limit the available parameter space for η , and couldbe interpreted in terms of corresponding limits on the maximum allowed value of η . In the current paper, we have analyzed the particle-antiparticle symmetry of thegravitationally (and electromagnetically) coupled Dirac equation and come to theconclusion that a symmetry exists, for the second-quantized formulation, which pre-cludes the existence particle-antiparticle symmetry breaking terms on the level ofDirac theory. In a nutshell, one might say the following: Just as much as the elec-tromagnetically coupled Dirac equation predicts that antiparticles have the oppositecharge as compared to particles (but otherwise behave exactly the same under elec-tromagnetic interactions), the gravitationally coupled Dirac equation predicts thatparticles and antiparticles follow exactly the same dynamics in curved space-time,i.e., with respect to gravitational fields (in particular, they have the same gravi-tational mass, and there is no sign change in the gravitational coupling). In thederivation of our theorem (33), we use the second-quantized Dirac formalism, in theLagrangian formulation. Our general result for the Dirac adjoint, communicated inSec. 2, paves the way for the Lagrangian of the gravitationally coupled field, and itsexplicit form is an essential ingredient of our considerations.Why is this interesting? Well, first, because the transformation of the gravitationalforce under the particle-to-antiparticle transformation has been discussed controver-sially in the literature [40, 41, 42, 43]. In Ref. [10], it was pointed out that the roleof the CPT transformation in gravity needs to be considered with care: It relatesthe fall of an apple on Earth to the fall of an anti-apple on anti-Earth, but not onEarth. The Dirac equation, colloquially speaking, applies to both apples as wellas anti-apples on Earth, i.e., to particles and antiparticles in the same space-timemetric. Second, our results have important consequences because one might haveotherwise speculated about the existence of tiny violations of the particle-antiparticlesymmetry, even on the level of the gravitationally coupled Dirac theory. For exam-ple, in Ref. [44], it was claimed that the Dirac Hamiltonian for a particle in a centralgravitational field, after a Foldy–Wouthuysen transformation which disentangles theparticle from the antiparticle degrees of freedom, contains the term [see the last termon the first line of the right-hand side of Eq. (31)] H ∼ − ~ c ~ Σ · ~g , ~ Σ = (cid:18) ~σ ~σ (cid:19) . (37)We here restore the factors ~ and c in order to facilitate the comparison to Ref. [44].The term proportional to ~ Σ · ~g , where ~g is the acceleration due to gravity, would10reak parity, because ~ Σ transforms as a pseudovector, while ~g transforms as a vectorunder parity. This aspect has given rise to discussion, based on the observation thatan initially parity-even Hamiltonian (in a central field) should not give rise to parity-breaking terms after a disentangling of the effective Hamiltonians for particles andantiparticles [45, 46].We should note that Ref. [44] was not the only place in the literature where theauthors speculated about the existence of P, and CP–violating terms obtained af-ter the identification of low-energy operators obtained from Dirac Hamiltonians ingravitational fields. E.g., in Eq. (46) of Ref. [47], spurious parity-violating, andCP-violating terms were obtained after a Foldy–Wouthuysen transformation; theseterms would of course also violate particle-antiparticle symmetry.In the context of the current discussion, the existence of terms proportional to ~ Σ · ~g , asgiven in Eq. (37), would also violate particle-antiparticle symmetry: This is becauseit lacks the universal prefactor β = γ , where β = (cid:18) × − × (cid:19) . (38)In fact, in the complete result (up to fourth order in the momenta) for the effectiveparticle-antiparticle Hamiltonian in a central field, given in Eq. (21) of Ref. [6], allterms have a common prefactor β . The common prefactor β implies that, after theapplication of the reinterpretation principle for antiparticles, the effective Hamil-tonians for particles and antiparticles in a central gravitational field (but withoutelectromagnetic coupling) are exactly the same, and ensures the particle-antiparticlesymmetry.The absence of such parity-violating (and particle-antiparticle symmetry breaking)terms has meanwhile been confirmed in remarks following Eq. (15) of Ref. [48], inthe text following Eq. (35) of Ref. [49], and also, in clarifying remarks given in thetext following Eq. (7.33) of Ref. [50]. Further clarifying analyses can be found inRef. [51] and in Ref. [52]. Related calculations have recently been considered in othercontexts [53, 54, 50]. The question of whether such parity- and particle-antiparticlesymmetry violating terms could exist in higher orders in the momentum expansionhas been answered negatively in Ref. [7], but only for a static central gravitationalfield, and in Ref. [55], still negatively, for combined static , central gravitational andelectrostatic fields.However, the question regarding the absence of particle-antiparticle symmetry break-ing terms for general, dynamic space-time backgrounds has not been answered con-clusively in the literature up to this point, to the best of our knowledge. Thishas been the task of the current paper. In particular, our results imply a no-gotheorem regarding the possible emergence of particle-antiparticle-symmetry break-ing gravitational, and combined electromagnetic-gravitational terms in general staticand dynamic curved-space-time backgrounds. Any speculation [47, 44] about there-emergence of such terms in a dynamic space-time background can thus be laidto rest. Concomitantly, we demonstrate that there are no “overlap” or “inter-ference” terms generated in the particle-antiparticle transformation, between the11auge groups, namely, the SO (1 ,
3) gauge group of the local Lorentz transforma-tions, and the U (1) gauge group of the electromagnetic theory. This result impliesboth progress and, unfortunately, some disappointment, because the emergence ofsuch terms would have been fascinating and would have opened up, quite possibly,interesting experimental opportunities. In our opinion, antimatter gravity experi-ments should be interpreted in terms of limits on CPT-violating parameters, suchas the η parameter introduced in Sec. 4. This may be somewhat less exciting thana “probe of the equivalence principle for antiparticles” but still, of utmost value forthe scientific community. Acknowledgments
The author acknowledges support from the National Science Foundation (GrantPHY–1710856) as well as insightful conversations with J. H. Noble.
A Sign Change of ψ ψ under Charge Conjugation
With the charge conjugation matrix C = i γ γ (superscripts denote Cartesian in-dices), and the Dirac adjoint ψ = ψ + γ , we have ψ C = C ψ T = i γ γ γ ψ ∗ = i γ ψ ∗ . (39)We recall that the γ (contravariant index, no square) matrix in the Dirac represen-tation matrix is γ = (cid:18) σ − σ (cid:19) , σ = (cid:18) − ii 0 (cid:19) , (cid:0) σ (cid:1) + = σ , (40)which implies that (cid:0) γ (cid:1) + = − γ . The Dirac adjoint of the charge conjugate is ψ C = (cid:0) ψ C (cid:1) + γ = ψ T ( − i) (cid:0) γ (cid:1) + γ = ψ T ( − i) ( − γ ) γ = ψ T i γ γ . (41)This leads to a verification of the sign flip of the mass terms in the gravitationallycoupled Lagrangian for antimatter, given in Eq. (27) [see also Eqs. (28) and (29)], ψ C ψ C = ( ψ T i γ ) γ (i γ ψ ∗ ) = − (i) ψ T ( γ ) γ ψ ∗ = − ψ T γ ψ ∗ = − ψ ψ . (42)Two useful identities (i) γ C + γ = C and (ii) C − = − C have been used in Sec. 3.These will be derived in the following. The explicit form of the γ matrix in theDirac representation implies that (cid:0) γ (cid:1) + = − γ . Based on this relation, we caneasily show that C + = (cid:0) i γ γ (cid:1) + = − i γ (cid:0) γ (cid:1) + = i γ γ = − i γ γ = − C . (43)The first identity γ C + γ = C can now be shown as follows, γ C + γ = γ (cid:2) − i γ γ (cid:3) γ = − i γ γ = i γ γ = C . (44)12urthermore, one has
C C + = C ( − C ) = i γ γ i γ γ = − (cid:0) γ (cid:1) = − ( − × ) = × , (45)so that C − = C + = − C , (46)which proves, in particular, that C − = − C . B General Considerations
A few illustrative remarks are in order. These concern the following questions: (i)
To which extent do gravitational and electrostatic interactions differ for relativisticparticles? This question is relevant because, in the nonrelativistic limit, in a centralfield, both interactions are described by potentials of the same functional form (“1 /R potentials”). (ii) Also, we should clarify why the integrals (28) and (29) represent thedominant terms in the evaluation of the Dirac particle energies, in the nonrelativisticlimit.After some rather deliberate and extensive considerations, one can show [8] that,up to corrections which combine momentum operators and potentials, the generalHamiltonian for a Dirac particle in a combined electric and gravitational field is H D = ~α · ~p + β { m (1 + φ G ) } + eφ C , (47)where φ G is the gravitational, and φ C is the electrostatic potential. Also, ~α is thevector of Dirac α matrices, ~p is the momentum operator, and β = γ is the Dirac β matrix. After a Foldy–Wouthuysen transformation [51], one sees that the gravi-tational interaction respects the particle-antiparticle symmetry, while the Coulombpotential does not, commensurate with the opposite sign of the charge for antipar-ticles. Question (i) as posed above can thus be answered with reference to the factthat, in leading approximation, the gravitational potential enters the Dirac equationas a scalar potential, modifying the mass term, while the electrostatic potential canbe added to the free Dirac Hamiltonian vecα · ~p + βm by covariant coupling [1].The second question posed above is now easy to answer: Namely, in the nonrela-tivistic limit, one has ~α · ~p → , (48)and furthermore, the gravitational and electrostatic potentials can be assumed tobe weak against the mass term, at least for non-extreme Coulomb fields [56]. Underthese assumptions, one has H D → βm , and the matrix element h ψ | H D | ψ i assumesthe form R d r ψ + ( ~r ) γ mψ ( ~r ) [see Eq. (28)]. References [1] C. Itzykson and J. B. Zuber,
Quantum Field Theory (McGraw-Hill, New York,1980). 132] C. D. Anderson, Phys. Rev. , 491 (1933).[3] G. E. Brown and D. G. Ravenhall, Proc. Roy. Soc. London, Ser. A , 552(1951).[4] R. Jauregui, C. F. Bunge, and E. Ley-Koo, Phys. Rev. A , 1781 (1997).[5] J. Maruani, J. Chin. Chem. Soc. , 33 (2016).[6] U. D. Jentschura and J. H. Noble, Phys. Rev. A , 022121 (2013).[7] U. D. Jentschura, Phys. Rev. A , 032101 (2013), [Erratum Phys. Rev. A ,069903(E) (2013)].[8] U. D. Jentschura, Phys. Rev. A , 032508 (2018).[9] U. D. Jentschura, Int. J. Mod. Phys. A , 1950180 (2019).[10] M. H. Holzscheiter, R. E. Brown, J. Camp, T. Darling, P. Dyer, D. B. Holtkamp,N. Jarmie, N. S. P. King, M. M. Schauer, S. Cornford, K. Hosea, R. A. Kenefick,M. Midzor, D. Oakley, R. Ristinen, and F. C. Witteborn, AIP Conf. Proc. ,573 (1991).[11] D. R. Brill and J. A. Wheeler, Rev. Mod. Phys. , 465 (1957).[12] V. Fock and D. Iwanenko, Z. Phys. , 798 (1929).[13] V. Fock, Z. Phys. , 261 (1929).[14] V. Fock and D. Ivanenko, C. R. Acad. Sci. Paris , 1470 (1929).[15] D. G. Boulware, Phys. Rev. D , 350 (1975).[16] M. Soffel, B. M¨uller, and W. Greiner, J. Phys. A , 551 (1977).[17] O. S. Ivanitskaya, Extended Lorentz transformations and their applications (inRussian) (Nauka i Technika, Minsk, USSR, 1969).[18] O. S. Ivanitskaya,
Lorentzian basis and gravitational effects in Einsteins theoryof gravity (in Russian) (Nauka i Technika, Minsk, USSR, 1969).[19] O. S. Ivanitskaya, N. V. Mitskievic, and Y. S. Vladimirov, in
Proceedings of the114th Symposium of the International Astronomical Union held in Leningrad,USSR, May 1985 , edited by J. Kovalevsky and V. A. Brumberg (Kluwer, Dor-drecht, 1985), pp. 177–186.[20] M. Bojowald,
Canonical Gravity and Applications (Cambridge University Press,Cambridge, 2011).[21] A. I. Akhiezer and V. B. Berestetskii,
Quantum Electrodynamics (Nauka,Moscow, 1969). 1422] M. E. Peskin and D. V. Schroeder,
An Introduction to Quantum Field Theory (Perseus, Cambridge, Massachusetts, 1995).[23] S. Gasiorowicz,
Elementarteilchenphysik (Bibliographisches Institut,Mannheim, 1975).[24] J. M. Jauch and F. Rohrlich,
The Theory of Photons and Electrons , 2 ed.(Springer, Heidelberg, 1980).[25] A. Lahiri and P. B. Pal,
Quantum Field Theory (Alpha Science, Oxford, UK,2011).[26] J. D. Bjorken and S. D. Drell,
Relativistic Quantum Mechanics (McGraw-Hill,New York, 1964).[27] J. D. Bjorken and S. D. Drell,
Relativistic Quantum Fields (McGraw-Hill, NewYork, 1965).[28] N. N. Bogoliubov, A. A. Logunov, and I. T. Todorov,
Introduction to AxiomaticQuantum Field Theory (W. A. Benjamin, Reading, Massachusetts, 1975).[29] H. Kleinert,
Particles and Quantum Fields (World Scientific, Singapore, 2016).[30] N. S. Mankoc Borstnik, Int. J. Theor. Phys. , 315 (2001).[31] N. S. Mankoc Borstnik and H. B. F. Nielsen, How to generate families of spinors ,preprint arXiv:hep-th/0303224.[32] N. S. Mankoc Borstnik, Phys. Rev. D , 065004 (2015).[33] N. S. Mankoc Borstnik and H. B. F. Nielsen, Progress in Physics , 1700046(2016).[34] N. Mankoc Borstnik, in Conference on New Physics at the Large Hadron Col-lider , edited by H. Fritzsch (World Scientific, Singapore, 2017), pp. 161–194.[35] T. Kaluza, Preussische Akademie der Wissenschaften (Berlin), Sitzungs-berichte, 966–972 (1921).[36] O. Klein, Z. Phys. A , 895 (1926).[37] M. J. Longo, Phys. Rev. Lett. , 173 (1988).[38] J. M. LoSecco, Phys. Rev. D , 3313 (1988).[39] J. H. Noble and U. D. Jentschura, Phys. Rev. A , 012101 (2015).[40] R. M. Santilli, Int. J. Mod. Phys. A , 2205 (1999).[41] M. Villata, Europhys. Lett. , 20001 (2011).[42] M. J. T. F. Cabbolet, Astrophys. Space Sci. , 5 (2011).1543] M. Villata, Astrophys. Space Sci. , 15 (2011).[44] Y. N. Obukhov, Phys. Rev. Lett. , 192 (2001).[45] N. Nicolaevici, Phys. Rev. Lett. , 068902 (2002).[46] Y. N. Obukhov, Phys. Rev. Lett. , 068903 (2002).[47] J. F. Donoghue and B. R. Holstein, Am. J. Phys. , 827 (1986).[48] A. J. Silenko and O. V. Teryaev, Phys. Rev. D , 064016 (2005).[49] A. J. Silenko, Phys. Rev. A , 032104 (2016).[50] Y. N. Obukhov, A. J. Silenko, and O. V. Teryaev, Phys. Rev. D , 105005(2017).[51] U. D. Jentschura and J. H. Noble, J. Phys. A , 045402 (2014).[52] M. V. Gorbatenko and V. P. Neznamov, Ann. Phys. (Berlin) , 195 (2014).[53] Y. N. Obukhov, A. J. Silenko, and O. V. Teryaev, Phys. Rev. D , 124068(2014).[54] Y. N. Obukhov, A. J. Silenko, and O. V. Teryaev, Phys. Rev. D , 044019(2016).[55] J. H. Noble and U. D. Jentschura, Phys. Rev. A , 032108 (2016).[56] P. J. Mohr, G. Plunien, and G. Soff, Phys. Rep.293