Gravitomagnetism and Spinor Quantum Mechanics

SLAC-PUB-15003

Gravitomagnetism and spinor quantum mechanics

Ronald J. Adler∗ 1.Gravity Probe B, Hansen Laboratory for Experimental Physics, Stanford University, Stanford CA, 94305 2.Department of Physics, San Francisco State University, San Francisco CA, 94132

Pisin Chen† 1. Leung Center for Cosmology and Particle Astrophysics & Department of Physics and Graduate Institute of Astrophysics, National Taiwan University, Taipei, Taiwan 10617 2. Kavli Institute for Particle Astrophysics and Cosmology, SLAC National Accelerator Laboratory, Menlo Park CA, 94025

Elisa Varani‡ Via Ponte Nuovo 24, 29014 Castell’Arquato (PC), Italy

We give a systematic treatment of a spin 1/2 particle in a combined electromagnetic field and a weak gravitational field that is produced by a slowly moving matter source. This paper continues previous work on a spin zero particle, but it is largely self-contained and may serve as an introduction to spinors in a Riemann space. The analysis is based on the Dirac equation expressed in generally covariant form and coupled minimally to the electromagnetic field. The restriction to a slowly moving matter source, such as the earth, allows us to describe the gravitational field by a gravitoelectric (Newtonian) potential and a gravitomagnetic (frame-dragging) vector potential, the existence of which has recently been experimentally verified. Our main interest is the coupling of the orbital and spin angular momenta of the particle to the gravitomagnetic field. Specifically we calculate

the gravitational gyromagnetic ratio as g g = 1 ; this is to be compared with the electromagnetic gyromagnetic ratio of ge = 2 for a Dirac electron.

PACS numbers: 03.65.-w, 03.65.Pm, 04.25.Nx, 04.80.Cc

1. INTRODUCTION oretical work has been done on quantum fields in classical background spaces, the most well known being related to Hawking radiation from black holes [1, 9–11]. However Classical systems in external gravitational fields have it is important to keep in mind that Hawking radiation been studied for centuries, and recently the existence of has never been observed. the gravitomagnetic (or frame-dragging) field caused by the earth’s rotation has been observed by the Gravity Interesting experimental work has also been done on Probe B (GPB) satellite [1–4]. GPB verified the predic- quantum systems in the earth’s gravitational field, such tion of general relativity for the gravitomagnetic preces- as neutrons interacting with the earths Newtonian field sion of a gyroscope in earth orbit (42 mas/yr) to better and atom interferometer experiments aimed at accurately than 20% [5]. Previously, observations of the LAGEOS testing the equivalence principle and other subtle gen- satellites also indicated the existence of the gravitomag- eral relativistic effects [12–14]. There has been some dis- netic interaction via its effect on the satellite orbits [5, 6]. cussion of attempts to see gravitomagnetic effects with Analysis of the LAGEOS data involves modeling classi- these devices but such experiments would be quite diffi- cal effects to very high accuracy in order to extract the cult due to the small size of the effects and the similarity gravitomagnetic effect, and the accuracy of the results to classical effects of rotation; this is to be expected since has been questioned by some authors [7]. Analysis of gravitomagnetism manifests itself in a way that is quite the GPB data also requires highly accurate modeling of similar to rotation, hence the appellation “frame drag- classical effects [5]. ging.” Laboratory detection of gravitomagnetic effects on a quantum system would clearly be of fundamental While gravitomagnetic effects are generally quite small interest. in the solar system it is widely believed that they may play a large role in jets from active galactic nuclei, so In this work we give a systematic treatment of a spin their experimental verification is of more than theoretical 1/2 particle in a combined electromagnetic field and weak interest [8]. gravitational field; this continues the work of reference At the other end of the interest spectrum extensive the- [15]. We describe the particle with the generally covariant Dirac equation in a Riemann space, minimally coupled to the electromagnetic field in the standard gauge invariant way [16, 17]. The weak gravitational field is ∗Electronic address: [email protected] naturally treated according to linearized general relativ- †Electronic address: [email protected] ity theory, and we also assume a slowly moving matter ‡Electronic address: [email protected] source, such as the earth [18–20]. Within this approxi-

Work supported in part by US Department of Energy contract DE-AC02-76SF00515. 2 mation the gravitational field is described by a gravito- 2. THE GRAVITOELECTROMAGNETIC (GEM) electric (or Newtonian) potential and a gravitomagnetic APPROXIMATION (or frame-dragging) vector potential, and the field equations are quite analogous to those of classical electromag- In previous work we discussed linearized general rela- netism. We thus refer to it as the gravitoelectromagnetic tivity theory for slowly moving matter sources like the (GEM) approximation. Our special emphasis throughout earth [15, 19, 20]. Here we summarize the results very this paper is on the gravitomagnetic interaction. briefly. The metric may be written as the Lorentz metric The paper is organized as follows. After brief review plus a small perturbation, comments on the GEM approximation (section 2) and the Dirac equation in flat space (section 3) we give a detailed gµν = ηµν + hµν . (2.1) discussion of generally covariant spinor theory and the Dirac equation, using the standard approach based on We use coordinate freedom to impose the Lorentz gauge tetrads (sections 4 and 5). We then obtain the limit of condition the Dirac Lagrangian and the Dirac equation for a weak 1 (h − η h)|ν = 0, (2.2) gravitational field and discuss its interpretation in terms µν 2 µν of an energy-momentum tensor (section 6). Our discussion of generally covariant spinors and where the single slash denotes an ordinary derivative. the generally covariant Dirac equation is largely self- Then the field equations of general relativity tell us that contained, and may serve as an introduction to the sub- the metric perturbation may be written as ject for uninitiated readers. In section 6 we also observe  1 2 3  that the non-geometric or “flat space gravity” approach 2φ h h h h1 2φ 0 0 of Feynman, Weinberg and others does not appear to be h =   , h = 2φ, h = hk, µν  h2 0 2φ 0  00 0k completely equivalent to linearized general relativity the- 3 ory in its coupling to spin [21]. We have not found this h 0 0 2φ discussed in the literature. (2.3) Using the weak gravitational field results we then ob- where φ is the Newtonian or gravitoelectric potential and tain the non-relativistic limit of the theory (section 7). hk ↔ ~h is the gravitomagnetic potential. For slowly mov- We do this by integrating the interaction Lagrangian to ing sources the field equations and the Lorentz condition obtain the interaction energy of the spinor particle with become the electromagnetic and the GEM fields, and from that obtain the non-relativistic interaction energies. This al- ˙ ∇2φ = 4πGρ, ∇2hj = −16πGρvj, 4φ˙ − ∇ · ~h = 0, ~h = 0, lows us to read off, in a simple and intuitive way, the (2.4) interaction terms that one could use in a non-relativistic Hamiltonian treatment. In particular we obtain (section where ρ is the source mass-energy density and vj is its 8) the usual anomalous g-factor of the electron g = 2 and e velocity. the analogous result for the gravitomagnetic g-factor of The physical fields, which exert forces on particles, are a spinor, which is g = 1. g the gravitoelectric (or Newtonian) field and the gravit- Section 8 also contains brief comments on the numeri- omagnetic (or frame-dragging) field, which are defined cal value of some interesting and conceivably observable by quantities such as the precession of a spinning particle in the earth’s gravitomagnetic field and its relation to ~g = ∇φ, Ω~ = ∇ × ~h. (2.5) the precession of a macroscopic gyroscope; such precession appears to be universal for bodies with angular mo- We call this equation system the gravitoelectromagnetic mentum. The phase shift in an atom interferometer is or GEM limit because of its close similarity to classical also mentioned as an experiment that could, in principle, electromagnetism. show the existence of the gravitomagnetic field. Lastly it is worth noting what we do not do in this paper. We study the effect of the gravitational field on 3. FLAT SPACE DIRAC EQUATION AND THE a quantum mechanical spinor but not the effect of the NON-RELATIVISTIC LIMIT spinor on the gravitational field; thus the work does not relate to quantum gravity or quantum spacetime [22]. In this section we discuss the Dirac equation in the Similarly we do not consider torsion, in which the affine flat space of special relativity and recast it into a connections have an anti-symmetric part and are not Schroedinger equation form (SEF), which provides one equal to the Christoffel symbols. Torsion does not prove convenient way to obtain the non-relativistic limit [16]. necessary in our discussion, but some authors believe it The SEF is exact and involves only the upper two com- is necessary in describing the effects of spin on gravity ponents of the spinor wave function - the relevant compo- [23]. nents for positive energy solutions in the non-relativistic 3

−1 limit. One reason for doing this is to serve as a ba- The inverse operator (2m−V +i∂t) may be defined by sis of comparison for the alternative method we will use its expansion in the time derivative, as discussed in Ap- in section 6 when we discuss gravitational interactions. pendix A. Note that Eq. (3.7a) is an exact equation for Throughout this section γµ denotes the flat space Dirac Ψ, although it is of infinite order in the time derivative. matrices [16, 24]. For the special case of a free particle the operator fac- The Dirac Lagrangian and the Euler-Lagrange equa- tors on the right side of (3.7a) commute and it becomes tions that follow from it are simply ¯ µ ~ ¯ µ←− ¯ µ −1 2 L = aψ(iγ ∂µ − m)ψ + bψ(−iγ ∂ µ − m)ψ − eAµψγ ψ, i∂tψ = (i∂t + 2m) ~p ψ. (3.8) (3.1a) However the operators on the right side of (3.7a) will not µ µ ¯ µ←− ¯ µ (iγ ∂µ − m)ψ = eAµγ ψ, ψ(−iγ ∂ µm) = eAµψγ . in general commute unless the field Aµ is constant. (3.1b) In a low velocity system the time variations of Ψ and V are associated with non-relativistic energies, which are The spinor and its adjoint are considered independent in much less than the rest energy m, so we may approximate obtaining (3.1b). The constants a and b are arbitrary, (3.7a) by so long as a + b 6= 0 . The γµ obey the flat space Dirac algebra, (~σ · Π)~ 2 i∂tΨ = V ψ + Ψ. (3.9) {γµ, γα} = 2ηµν I. (3.1c) 2m The adjoint spinor is assumed to be related to the spinor This is the Schroedinger equation for spin 1/2 parti- by a linear metric relation, ψ¯ = ψ†M where M is to be cles, often called the Pauli equation. The Pauli equation determined; consistency of the equations (3.1b) is then shows clearly how the spin and orbital angular momen- assured if M obeys tum interact with the magnetic field. Pauli spin matrix algebra leads to an illuminating form for (3.9): to lowest † M −1γˆµ M =γ ˆµ,M −1 = M =γ ˆ0, ψ¯ = ψ†γ0. (3.2) order in e, Π~ 2 eB~ · σ Eq. (3.2) is easy to verify for the choice of gamma ma- i∂ Ψ = V Ψ + Ψ − Ψ trices given below in (3.4). t 2m 2m The Hamiltonian form of the Dirac equation will be ~p2 eA~ · ~p eB~ · σ = V Ψ + Ψ − Ψ − Ψ, (3.10) useful for studying interaction energies in this section. It 2m m 2m is gotten by multiplying (3.1) by γ0 to obtain where we have used the Lorentz gauge in which ∇ · A~ = ~ 0 0 k i∂tψ = βmψ + V + ~α · Πψ, β ≡ γ , α ≡ γ γ , ~p ≡ −i∇. −A˙ 0 and assumed the Coulombic A0 has negligible time (3.3) dependence. The gyromagnetic ratio or g-factor of a particle or system is defined in terms of its magnetic moment Pauli’s choice of gamma matrices is natural for our later ~ ~ discussion of the non-relativistic limit, ~µ and angular momentum J by ~µ = ge(e/2m)J ; thus, from (3.10), the fact that the energy is −~µ · B~ , and the i 0 I 0 i 0 σ 0 σ electron spin of S~ = σ/2 it is evident that the electron β = γ = , γ = i , ~α ≡ . 0 I −σ 0 σ 0 g-factor is ge = 2. (3.4) The relative coupling of the spin and orbital magnetic moments is made most clear if we consider a magnetic Next we break the 4-component wave function ψ into field that is approximately constant over the size of the two 2-component Pauli spinor wave functions and also system, in which case we can choose A~ = (B~ × ~r)/2 and factor out the time dependence due to the rest mass by find from (3.10) substituting 2 ~ Ψ ~p eB ~ ~ ψ = e−imt , (3.5) i∂tΨ = V Ψ + Ψ − (2S + L)Ψ, ϕ 2m 2m S~ = σ/2, L~ = ~r × ~p. (3.11) which leads to the coupled equations, That is g = 2 for the electron spin and g = 1 for the ~ ~ e e i∂tΨ = V Ψ + (~σ · Π)ϕ, i∂tφ + 2mϕ − V ϕ = (~σ · P i)Ψ. orbital angular momentum. (3.6) Equation (3.7a) may be expanded to higher order to study such things as hyperfine structure and relativistic We are interested in Ψ so we solve for ϕ , and obtain corrections in the hydrogen atom spectrum [25]. That is symbolically, ~ 2 ~ ~ ~ −1 ~ (~σ · Π) (~σ · Π)(i∂t − V )(~σ · Π) i∂tΨ = V Ψ + (~σ · Π)(2m − V + i∂t) (~σ · Π)Ψ, (3.7a) i∂ Ψ = V Ψ + Ψ − Ψ, t 2m 4m2 −1 ϕ = (2m − V + i∂t) (~σ · Π)Ψ~ . (3.7b) (3.12) 4

However an important problem and caveat is that the vector we define a rule for transplanting a spinor from x wave function Ψ in (3.12) is only the upper half of the to a nearby point x + dx , Dirac wave function, so the quantity that must be nor- 2 2 2 ψ∗(x + dx) = ψ(x) − Γ ψ(x)dxµ. (4.5) malized is |Ψ| +|ϕ| rather than |ψs| for a Schroedinger µ or Pauli wave function ψ . Thus to insure Hermiticity s The matrices Γ are variously called spin connections, and conserve probability one must renormalize the wave a affine spin connections, or Fock-Ivanenko coefficients. function as discussed in detail in ref. [25]. It is for this The covariant derivative is then defined in terms of the reason that we will adopt an alternative and conceptually difference between the value of the spinor and the value simpler approach to the non-relativistic limit in section it would have if transplanted to the nearby point. That 7. is

ν ν ν ψ(x)||ν dx = [ψ(x) + ψ(x)|ν dx ] − [ψ(x) − Γν (x)ψ(x)dx ] 4. GENERALLY COVARIANT SPINOR ν THEORY = [ψ(x)|ν + Γν (x)ψ(x)]dx ,

ψ||ν = ψ|ν + Γν ψ = (∂ν + Γν )ψ ≡ Dν ψ. (4.6) The gravitational interaction of a spinor may be obtained most easily by making the Dirac Lagrangian (3.1a) Here the double slash denotes a covariant derivative. and Dirac equation (3.1b) generally covariant. To do this Since the spinor covariant derivative must transform as a we adopt the standard approach of using a tetrad of basis vector under coordinate transformations and as a spinor vectors in order to relate the generally covariant theory under Lorentz transformations of the tetrad basis, we to the special relativistic theory in Lorentz coordinates have [14, 17]. This is a most natural, almost inevitable, ap- ∂xν ψ0 = Sψ , (4.7) proach since Dirac spinors transform by the lowest di- ||µ ∂x0µ ||ν mensional representation S of the Lorentz group; that is ψ0 = Sψ . It follows from (4.6) and (4.7) that the spin connections Two properties of the Dirac Lagrangian and Dirac must transform according to equation must be modified to obtain a generally covari- ν 0 ∂x −1 −1 ant theory: the Dirac algebra in (3.2) must be made Γν = 0µ [SΓν S − S|ν S ]. (4.8) covariant and the derivative of the spinor in (3.1) must ∂x be made into a covariant derivative. We will discuss both The transformation (4.8) is formally similar to that of the in detail. affine connections used for vector covariant derivatives. The Dirac algebra (3.1c) is easily made covariant by re- The covariant derivative of an adjoint spinor follows placing the Lorentz metric ηµν by the Riemannian metric easily from that of a spinor in (4.6); we ask that the gµν , inner product ψχ¯ of a spinor χ and an adjoint spinor ψ¯ ¯ be a scalar and thus have a covariant derivative (ψχ)||µ {γµ, γα} = 2gµν I. (4.1) ¯ equal to the ordinary derivative (ψχ)|µ, and we also ask µ that the product rule hold for both the ordinary and the A set of γ matrices that satisfy (4.1) is easily con- covariant derivatives. The result is structed by using a set of constantγ ˆb that satisfies the µ ¯ ¯ ¯ special relativistic relation (3.2) and a tetrad field eb nor- ψ||µ = ψ|µ − ψΓµ. (4.9) malized by the usual tetrad relations The same idea leads to the covariant derivative of a µ ν αβ α β cd gamma matrix, with only a bit more algebra; that is we eb ea gµν = ηab, g = ec ed η . (4.2) ¯ µ ask that the expression (ψγ χ)||α be a second rank ten- Here the Greek indices label components of the tetrad sor and that it obey the product rule of differentiation, vectors and Latin indices label the vectors. In terms of and find from (4.6) and (4.9) a convenient set of constant Dirac matricesγ ˆb, such as µ µ those in (3.4), we define the γ by γµ = γµ + + [Γ , γµ]. (4.10) ||ω |ω ωσ ω γµ = eµγˆb. (4.3) b This expression plays an important role in obtaining the It then follows from (3.1c) and (4.2) that the γµ satisfy spin connections in the next section.

µ ν µ ν b a µ ν ab µν {γ , γ } = eb ea {γˆ , γˆ } = eb ea 2η I = 2g I. (4.4) 5. COVARIANT DIRAC LAGRANGIAN AND The covariant derivative of a spinor is deﬁned so as to DIRAC EQUATION transform as a vector under general coordinate transformations and as a spinor under Lorentz transformation of In this section we give a covariant Lagrangian and ob- the tetrad basis. As with the covariant derivative of a tain the covariant Dirac equation. In the process we get 5

µ a relation between the spinor and its adjoint (i.e. a spin This guarantees that the divergence vanishes, γ ||µ = 0, metric) and evaluate the spin connections. as we have already mentioned. However it is a stronger The choice of a covariant Dirac Lagrangian L, and its demand analogous to the standard demand in general rel- associated Lagrangian density L, is rather obvious from ativity that the metric have a null covariant derivative, the flat space Lagrangian in (3.1), which forces the affine connections to be the Christoffel √ symbols. Note also that Γ is obviously arbitrary up to L = aψ¯(iγµψ − mψ) + b(−iψ¯ γµ − ψm¯ )ψ, L = gL. α ||µ ||µ a multiple of the identity, which we will suppress hence- (5.1) forth. µ Coupling to the electromagnetic field will be included To solve (5.7) we express γ in terms of flat space gam- b later. The γµ denotes the covariant Dirac matrices (4.3) masγ ˆ as in (4.3) and rewrite (5.7) as throughout this section. The Dirac equations for the µ b b µ spinor and the adjoint spinor follow directly as the Euler- eb||αγˆ + [Γα, γˆ ]eb = 0. (5.8) Lagrange equations of the Lagrangian density L with ψ and ψ¯ treated as independent variables, Multiplying this by the inverse tetrad matrix we get µ µ [Γ , γˆc] = −ec eµ γˆb. (5.9) (a + b)(iγ ψ||µ − mψ) + ibγ ||µψ = 0 (5.2a) α µ b||α (a + b)(ψ¯ iγµ + mψ) + iaψγ¯ µ = 0. (5.2b) ||µ ||µ We next note the well-known commutation relation on the sigma matrices, which are defined asσ âb ≡ For simplicity we assume that the spin connections, un- a b specified up to this point, may be chosen so that the di- (i/2)[ˆγ , γˆ ], vergence of γα that appears in (5.2) vanishes, γµ = 0. ||µ [ˆσab, γˆc] = 2i(ˆγaηbc − γˆbηac), (5.10) The covariant Dirac equation is then the obvious gen- eralization of the flat space equations (3.1). The spin From (5.10) it is evident that we should seek a solution connections will be obtained below. Also for simplicity that is proportional toσ âb times a product of the tetrad and symmetry we choose henceforth a = b = 1/2; this and its derivatives. It is easy to verify that the specific will prove convenient later. choice Next, as in flat space in section 3, we ask that there be a relation between the adjoint and the spinor, ψ¯ = i µ ab Γα = ebµe σˆ , (5.11) ψ†M, such that the two equations (5.2) are consistent. 4 a||α Manipulating (5.2a) we get for the adjoint, satisfies (5.9) and thus serves as the spin connection. ¯ µ ¯ −1 µ ¯˜ µ ¯ We thus have obtained a generally covariant theory in −iψ|µγ˜ − iψM |µ Mγ˜ − iψΓµγ˜ − ψm = 0, which the Lagrangian, the Dirac equations, the relation µ −1 µ† ˜ −1 † γ˜ ≡ M γ M, Γµ ≡ M Γµ M. (5.3) of the spinor to its adjoint, and the spin connections are generally covariant and consistent. We then compare (5.3) with (5.2b), written as Finally we include coupling to the electromagnetic field ¯ µ ¯ µ ¯ −iψ|µγ + iψΓµγ − ψm = 0, (5.4) in the usual minimal coupling way, that is by substituting iDµ → iDµ − eAµ; this gives the complete covariant and see that M must satisfy the following two equations Lagrangian 1 1 µ µ −1 µ† ¯ µ ¯ µ ¯ ¯ µ γ =γ ˜ = M γ M, (5.5a) L = ψ(iγ ψ||µ − mψ) + (−iψ||µγ − ψm)ψ − eAµψγ ψ. 2 2 ˜ −1 † −1 (5.12) −Γµ = Γµ = M Γµ M + M |µM. (5.5b) Eq. (5.5a) may be written in terms of flat spaceγ ˆb as We will study the weak gravitational field limit of this in the next section. µ b µ −1 b† eb γˆ = eb M γˆ M. (5.6)

† Thus it is obvious that we should askγ ˆb = M −1γˆb M, 6. LINEARIZED THEORY FOR WEAK which is the same as in the case of flat space and special GRAVITY relativity (3.2), so we may choose M −1 = M =γ ˆ0. Then the derivative of M is zero, and it is easy to verify that In this section we use the results of section 5 for covari- the choice M −1 = M =γ ˆ0 also satisfies (5.5b). ant spinor theory to work out the weak field linearized Our remaining task is to obtain specific spin connec- theory. This is done by setting up an appropriate tetrad tions Γα. To do this we make the natural demand that and using it to expand the Lagrangian (5.12) to lowest γµ have a null covariant derivative, so from (4.10) order in the metric perturbation. The result is that there are three interaction terms in the Lagrangian, one asso- µ µ µ β µ ciated with the spin coefficients and the second with the γ||α = γ|α + γ + [Γα, γ ] = 0. (5.7) αβ alteration in the γµ caused by gravity. Remarkably the 6

first vanishes in the linearized theory, while the second Thus the interaction term containing the spin connec- corresponds to an interaction via the energy momentum tions in (6.5) vanishes, which is a remarkable simplifi- tensor, as intuition should suggest. The third term is a cation. It should be stressed that this is only true to cross term between the weak gravity and electromagnetic first order, and the spin connections will generally be of fields. interest in the full theory. In a space with a nearly Lorentz metric (2.1) it is natu- There remains in the Lagrangian (6.5) only interac- ral to choose a tetrad that lies nearly along the coordinate tions due to the modification of theγ ˆµ by gravity in axes, (6.2); L may now be written as

eµ = δµ + wµ, eb = δb − wb , (6.1) 1 1 a a a ν ν ν L = ψ¯(iγψˆ − mψ) + (−iψ¯ γˆ − ψm¯ )ψ − eA ψ¯γˆµψ 2 |µ 2 |µ µ where wµ is a small quantity to be determined. From a 1 1 1 the fundamental tetrad relation (4.2) it follows that we − hµ [ ψ¯γˆα(iψ − eA ψ) − (iψ¯ + eA ψ¯)ˆγαψ] 2 α 2 |µ µ 2 |µ µ should choose a symmetric wµν = −(1/2)hµν and thus (6.8) have a tetrad and γµ matrices given by

µ µ µ The quantity in brackets in (6.8) is the appropriately ea = δa − (1/2)ha , a µ µ µ a µ µ a symmetrized energy-momentum tensor T µ for the Dirac γ = [δa − (1/2)ha ]ˆγ =γ ˆ − (1/2)ha γˆ . (6.2) field interacting with the electromagnetic field; that Since Greek tensor indices and Latin tetrad indices are is, the gravitational interaction Lagrangian may be ex- intimately mixed in the linearized theory we will not dis- pressed as tinguish between them in this section. 1 µ 1 ¯ α 1 ¯ ¯ α To evaluate the spin connections (5.11) with the tetrad LIG = − h α[ ψγˆ (iψ|µ − eAµψ) − (iψ|µ + eAµψ)ˆγ ψ] (6.2) we need the Christoffel symbols and the covariant 2 2 2 1 µα derivative of the tetrad to first order in hµν , = − h T . 2 µα ν (6.9) = (1/2)(h ν + h ν − h |ν ), µω ω |µ µ |ω µω The energy momentum tensor is discussed further in Ap- ν ν |ν ea||µ = (1/2)(hµ |a − hµa ). (6.3) pendix B. The interaction (6.9) consists of the inner product of From (5.11), (6.2) and (6.3) we obtain the spin connec- the field hµν with the conserved energy-momentum ten- tions, sor T µν ; this coupling is in close analogy with the elec- i i tromagnetic coupling between the field Aµ and the con- ν ab ∼ ab µ µ Γµ = ebν e σˆ = hµb|aσˆ . (6.4) served current j = eψγ¯ ψ in (6.8). Feynman has em- 4 a||µ 4 phasized this analogy and developed a complete “flat Thus the Dirac Lagrangian (5.12) becomes, space” gravitational theory, with gravity treated as an “ordinary” two index (spin 2) field and formulated by 1 ¯ µ 1 ¯ µ ¯ ¯ µ L = ψ(iγ ψ|µ − mψ) + (−iψ|µγ − ψm)ψ − eAµψγ ψ analogy with electromagnetism, at least to lowest order 2 2 [21]. The geometric interpretation of gravity is thereby i ¯ i µ ¯ α ¯ α 1 µ ¯ α suppressed or ignored. Weinberg has similarly stressed + ψ{γ,ˆ Γµ}ψ − h α[ψγˆ ψ|µ − ψ|µγˆ ψ] + h αAµψγˆ ψ, 2 4 2 that the geometric interpretation of gravity is not es- (6.5) sential [14, 21]. Schwinger also has used a similar and probably equivalent non-geometric methodology called with Γµ given in (6.4). The first line is the Dirac La- grangian in flat space (3.1a), and the other three terms source theory to obtain the standard results of general are gravitational interactions that we now address. relativity theory, including the precession of a gyroscope The first interaction term in the second line of due to the gravitomagnetic field [26]. However there is a (6.5), due to the spin connections, contains the anti- problem with relating the geometric and non-geometric µ viewpoints, in that the Euler-Lagrange field equations commutator {γˆ , Γµ}. With the use of the symmetry √ are based on the Lagrangian density L gL =∼ (1+h/2)L of hµν , the Dirac algebra (3.1c), and the operator identity [AB, C] = A{B,C} − {A, C}B it is straightforward and not the Lagrangian L, so there is an additional in- to verify the following two expressions, teraction term (h/2)L in the geometric theory that is not present in the non-geometric theory; the equivalence of µ ab a b hµb|aγˆ σˆ = i(h b|a − h|b)ˆγ , the Feynman approach to the linearized geometric approach is thus spoiled whenever the additional term does h σâbγˆµ = i(h − ha )ˆγb, (6.6) µb|a |b b|a not vanish. and thereby see that The difference between the Dirac equation per our geometric development and that which one would obtain µ i µ ab from the non-geometric approach is easy to see. The {γˆ , Γ } = h {γˆ , σˆ } = 0. (6.7) √ µ 4 µb|a Dirac equation that follows from (6.8) with L = gL =∼ 7

(1 + h/2)L is For the Dirac ψ we use a convenient device, an expansion in terms of free positive energy Dirac wave functions on µ γ (iψ|µ − eAµ) − mψ the mass shell. That is

1 µ |ν ν 1 µ ν = hµν γˆ (iψ − eA ψ) + (h − h|ν )iγˆ ψ. 2 4 ν|µ X Z d3p ψ = f(p, s)[eipα xαu(p, s)], (6.10) (2π)3 s=1,2 2 0 2 2 2 The last term on the right containing h|ν would not be E = (p ) = p + m . (7.2) present in the non-geometric approach. This will be discussed further in section 7. In summary of this section, the Lagrangian (6.8) con- The positive energy wave functions do not form a com- tains the interaction of the Dirac field with the electro- plete set, but the approximation (7.2) should be quite magnetic field to all orders and the interaction with the good for distances much larger than the Compton wave- gravitational field only to lowest order; (6.10) is the cor- length, ~/m; (7.2) is our fundamental assumption. A key responding Dirac equation. We will discuss the interac- idea in the calculation is to express the Dirac 4-spinor tion energies further in the following section in which we u(p, s) in terms of a Pauli 2-spinor χs [27], consider the non-relativistic or low velocity limit of the theory. r α α E + M I −ipαx −ipαx e u(p, s) = e ~σ·~p χs. (7.3) 2m E+M 7. NON-RELATIVISTIC LIMIT Correspondingly we express the non-relativistic Pauli We wish to use the results of the previous sections to wave function as obtain a non-relativistic limit of the theory and calcu- Z 3 late in a simple way some interesting properties of a spin X d p α Ψ = f(p, s)eipαx χ . (7.4) 1/2 particle such as the electromagnetic g-factor and its (2π)3 s gravitomagnetic analogue. The most familiar approach s=1,2 to this problem is to work with the upper two components of the Dirac wave function as we did in section 3, In terms of the above expressions (7.2) and (7.3) the in- and take the non-relativistic limit [16, 25]. However the teraction energy (7.1b) is alternative approach we use in this section is conceptually simpler and avoids the problems of renormalization X Z d3p d3p0 and Hermiticity that occur in the approach of sec. 3. The ∆E = f ∗(p0, s0)f(p, s) EM (2π)3 (2π)3 basic idea is to integrate the interaction Lagrangian over s,s0=1,2 3-space to get the interaction energy, then put the en- Z 0 α 3 i(p −pα)x 0 0 µ ergy expression with Dirac 4-spinor wave functions, into [e d xe α u¯(p , s )γ u(p, s)Aµ]. a form using Pauli 2-spinor wave functions, all in the low (7.5) velocity limit. In this section we will always work in nearly flat space µ with Lorentz coordinates; the Dirac γ will be those of The bracket in (7.5) corresponds to scattering of a free flat space and no hat will be used. Moreover for simplicity Dirac spinor by an external field, which is equivalent we will work in the Lorentz gauge for both the electro- to scattering by an infinitely heavy source particle. It magnetic and GEM fields, and take both the Coulomb 0 contains all the information about the spin interaction potential A and the Newtonian potential φ to have neg- and corresponds to the diagram in fig. 7.1: the parti- ˙ 0 ~ ligible time dependence; that is A = −∇ · A = 0 and cle leaves the wave function blob with 3-momentum ~p, 4φ˙ = ∇ · ~h = 0. This is quite appropriate, for exam- scatters from the external field via the QED vertex am- ple, for electromagnetic interactions in atoms and GEM plitude into momentum p~0, and then reenters the wave interactions on the earth. function blob. The electron remains on the mass shell, To illustrate the method we first consider only the elec- corresponding to zero energy transfer, which is consistent tromagnetic interaction in flat space; the results will be with a non-relativistic wave function. We denote the 4- the same as those in section 3, in particular g = 2. The 0 e momentum transfer by qµ = pµ − pµ, with q0 = 0 . The interaction Lagrangian and the interaction energy are, magnitude of the allowed 3-momentum transfer ~q is lim- from (6.8), ited by the width of the function f(p, s) in momentum space. ¯ µ µ LIEM = −eAµ(ψγ ψ) = −Aµj , (7.1a) Z It is now straightforward to calculate the bracket in 3 (7.5). We split it into 2 parts, µ = 0 for the electric ∆EEM = − LIEM d x. (7.1b) interaction and µ = j for the magnetic interaction. For 8

have

Z α 3 iqαx 0 0 j e d xAje u¯(p , s )γ u(p, s) Z α E + m 3 iqαx † = e d xe A χ 0 j 2m s ~σ · ~p0 0 σj I I, j ~σ·~p χs E + m σ 0 E+m Z α e f (p,s) f (p′ , s′ ) 3 iqαx † j 0 j = d xe Aj χs0 [σ ~σ · ~p + ~σ · ~p σ ]χs Aµ 2m Z α e 3 i~qαx † ~ ~ ~ = − d xe χ 0 [2~p · A + ~q · A + i~q × A · ~σ]χ . 2m s s (7.8) € € € We then replace ~q → −i∇ as discussed above and see that the second term in the bracket vanishes in a gauge FIG. 1: The electron in the wave function scatters from the with ∇ · A~ = 0, and we are left with ﬁeld and back into the wave function.

Z α 3 iqαx 0 0 j e d xAje u¯(p , s )γ u(p, s) the electric part we have Z α e 3 iqαx † ~ ~ = − d xe χ 0 [2~p · A + ∇ × A · ~σ]χ 2m s s Z α 3 iqαx 0 0 0 e d xA0e u¯(p , s )γ u(p, s) Z α h e e i 3 iqαx ~ † ~ † = − d xe ~p · A(χs0 χs) + B · (χs0 ~σχs) . 0 m 2m Z α E + m ~σ · ~p I 3 iqαx † (7.9) = e d xe A0 χs0 I, ~σ·~p χs 2m E + M E+m Z α Finally we combine (7.7) and (7.9) and substitute into = e d3xeiqαx (7.5) to obtain, to order 1/m, E ~q · ~p i~q × ~p · ~σ † Z 3 3 0 A0χ 0 + + χs. X d p d p s m 2m(E + m) 2m(E + m) ∆E = f ∗(p0, s0)f(p, s) EM (2π)3 (2π)3 (7.6) s,s0=1,2 Z 3 −i(p~0−~p)·~x † e ~ e ~ d xe χ 0 [eA − ~p · A B · ~σ]χ The first term in the bracket in (7.6) is the obvious charge s 0 m 2m s coupling to the Coulomb field. The second and third Z 3 † e ~ e ~ terms may be simplified. First, because there is no energy = d xΨ [eA0 − ~p · A − B · ~σ]Ψ. (7.10) 2 m 2m transferred ~p2 = p~0 , from which it follows that ~p · ~q = −~q2/2. Secondly the vector ~q multiplying the exponential This is the same result that we discussed in section 3, so may be replaced by i∇ operating on the exponential, we have thus verified that our present approach repro- after which integration by parts allows us to replace it duces the usual result for the electron g factor, ge = 2. with −i∇ operating on the function A0; that is we may We now work out the non-relativistic limit of the grav- replace ~qA0 → −i∇A0. Thus the second term vanishes itational interaction in (6.8), following the same proce- 2 since ∇ A0 = 0 in a charge free region for the Lorentz dure as for the electromagnetic interaction; we will not gauge. What remains is, to order 1/m2, include the product of the electromagnetic and gravitational fields, that is the cross term in (6.8). The algebra Z α is a bit lengthier but equally straightforward. As with 3 iqαx 0 0 0 e d xA0e u¯(p , s )γ u(p, s) the Lagrangian and energy for the electromagnetic case in (7.1) we have for the gravitational case Z α h e i 3 iqαx † † = d e e(χs0 χs) + 2 ∇φc × ~p · (χs0 ~σχs) . 4m 1 Z (7.7) L = − h T µν , ∆E = − L d3x, (7.11) IG 2 µν G IG

The second term in (7.7) is clearly a nonlocal ﬁne struc- where T µν is given in (6.9). It is convenient to write T µν ture correction, which we mentioned in sec. 4 and which in close analogy with the electromagnetic current, as will not concern us further in the present work [25]. 1 ←→ The µ = j magnetic part of the interaction (7.5) is T µα = ψ¯γˆα( i ∂ µ)ψ. (7.12) handled in exactly the same way as the electric part. We 2 9

Note the relation between the electromagnetic and the Note that the term ~q ·~h will vanish by gauge choice, just gravitational interactions, as the ~q · A~ term vanished for the electromagnetic case. Then, using the same manipulations as previously on the µ µ i ←→ν Aµ ↔ hµν /2, γ ↔ γ ( ∂ ). (7.13) spin products we reduce this to 2 Z Then ∆EG is, in analogy with (7.5), 3 i(p0 −p )xα 0 0 0 j j j [ d xe α α u¯(p , s ){γ (p + q /2) + Eγ } Z d3p d3p0 X ∗ 0 0 j ∆EG = f (p , s )f(p, s) u(p, s)(h /2)] (2π)3 (2π)3 s,s0=1,2 Z 0 α † 1 3 i(pα−pα)x ~ ~ Z 0 α = [ d xe χs0 {~p · h + ∇×h · ~σ}χ], 3 i(p −pα)x 0 0 µ ν ν [ d xe α u¯(p , s )γ (p + q /2)u(p, s)(hµν /2)]. 4 (7.17) (7.14) where we have neglected terms of higher order, that is As with the electromagnetism calculation we split the 1/m2. Finally we combine (7.15) and (7.17) to obtain gravitational interaction into two parts, the gravitoelec- the total energy tric for h00 = hii = 2φ and the gravitomagnetic for h = h = hj. The gravitoelectric part of the bracket 0j j0 X Z d3p d3p0 in (7.14) involves the same spin products as encountered ∆E = f ∗(p0, s0)f(p, s) G (2π)3 (2π)3 with the electromagnetic calculation in (7.7) and (7.9), s,s0=1,2 2 Z and after some algebra we obtain, to order 1/m , 0 α † 1 3 i(pα−pα)x ~ ~ [ d xe χ 0 (mφ + ~p · h + ∇ × h · ~σ)χ] Z j s 4 3 i(p0 −p )xα 0 0 0 j q j [ d xe α α u¯(p , s ){γ E + (p + )γ }u(p, s)φ] Z 1 2 = d3xΨ†(mφ + ~p · ~h + Ω~ · ~σ)Ψ. (7.18) 2 4 Z 0 α E E 3 i(pα−pα)x † = d xe χs0 [ φ + 2 ∇φ × ~p · ~σ m 4m (Recall that the gravitomagnetic field is Ω~ = ∇ × ~h.) ~p2 1 This is the main result of this section and is consistent + φ + ∇φ × ~p · ~σ ]χ m 2m s with the result of ref. [15] for a scalar particle. 2 Finally we note that since we have expanded the wave Z 0 α 2~p 3 3 i(pα−pα)x † = d xe mχ 0 [(1 + )φ + ∇φ × ~p · ~σ]χs. function in terms of a free Dirac particle on the mass s m2 4m2 shell (7.2) the free Dirac Lagrangian is zero and the extra (7.15) geometric interaction term (h/2)L discussed in section 6 A word is in order about the physical interpretation of vanishes. the gravitoelectric result (7.15). The term mφ is of course the expected Newtonian energy; the factor (1 + 2~p2/m2) occurs also in the analysis of a spin zero system in ref. 8. GRAVITOMAGNETIC PHYSICAL EFFECTS [15], and is approximately the Lorentz transformation factor between the potential in the lab frame and the The result (7.18) is to be compared with the analogous moving frame of the particle; thus (1 + 2~p2/m2)φ is the electromagnetic result (7.10). We see, of course, that Newtonian potential seen by the moving particle. The the Newtonian potential is the analog of the Coulomb last term in the bracket has the same form and is the potential eA0 and the gravitomagnetic potential is the gravitational analog of the fine structure term in the elec- analog of the vector potential according to tromagnetic energy (7.7), except of course for the different coefficient. We will not be concerned further with eA0 ↔ φ, (−e/m)A~ ↔ ~h. (8.1) the higher order terms in (7.15) and will henceforth keep only the lowest order term φ in the bracket. We also see that the coupling of the spin to the gravit- We turn finally to the gravitomagnetic part of the in- omagnetic field Ω~ is only half the analogous electromag- teraction (7.14), which is our main interest in this work. netic coupling. To make this most obvious we consider The gravitomagnetic part of the bracket, proportional to a gravitomagnetic field Ω~ that is approximately constant j h , is over the system so that we may choose ~h = (Ω~ × ~r)/2. Z 3 i(p0 −p )xα 0 0 0 j j j Then [ d xe α α u¯(p , s ){γ (p + q /2) + Eγ } Z 3 † 1 1 j ∆E = d xΨ (mφ + Ω~ × ~r · ~p + Ω~ · ~σ)Ψ u(p, s)(h /2)] G 2 4 Z 3 i(p0 −p )xα † 0 0 j j j Z 1 1 ~σ = [ d xe α α u (p , s ){(p + q /2)(h /2) = d3xΨ†(mφ + Ω~ · ~r × ~p + Ω~ · )Ψ 2 2 2 +E(hj/2)αj}u(p, s)]. Z 1 = d3xΨ†[mφ + Ω~ · (L~ + S~)]Ψ (8.2) (7.16) 2 10

Both orbital and spin angular momenta couple in the field and the frame-dragging or gravitomagnetic field same way to the gravitomagnetic field, so there is no take simple and intuitive forms. Detection of the small anomalous g factor for gravitomagnetism; that is gg = 1 gravitomagnetic effects in earth-based or satellite exper- for both orbital and spin angular momenta. iments is quite difficult, but such effects are expected to From the above correspondence it is clear that since be large and of great interest in astrophysical systems a magnetic moment due to orbital angular momentum, such as jets from active galactic nuclei and black holes. (e/2m)L~ , precesses at the Larmor frequency (eB/2m) in a magnetic field B, the gravitomagnetic moment due to both orbital and spin angular momenta will precess in a gravitomagnetic field Ω with frequency Ω, but in the op- Appendix A. THE INVERSE DIFFERENTIAL posite direction. Thus quantum precession should be the OPERATOR same as that observed in the classical gyroscope systems of the GPB satellite experiment [4]. It thus seems very We briefly study the type of differential operator that likely that the precession rate is universal for any angu- appears in (3.7) by solving the differential equation lar momentum system, whether the angular momentum is classical or quantum mechanical, orbital or spin. Af + ∂f = (A + ∂)f = F, f = f(x),F = F (x), For the surface of the earth the magnitude of the grav- (A.1) itomagnetic field is quite small, as estimated in ref. [15] The field and the associated quantum energy are of order where F (x) is a given function that may be expanded as a power series in the region of interest and A is a constant. −13 −28 Ω ≈ 10 rad/s, EΩ = ~Ω ≈ 10 eV. (8.3) The solution of the homogeneous equation is −Ax Experimental detection of such small quantum gravito- fh = Ce (C = arbitrary constant). (A.2) magnetic effects in an earth-based lab would obviously be difficult. Such an experiment might be performed The general solution of (A.1) is fh plus any particular with an atomic interferometer. The atomic beam could solution fp; for the particular solution we solve (A.1) be split into two components with angular momenta symbolically as, differing by ∆L = . Then, according to (8.2) the ~ 2 two components would have energies differing by about −1 1 ∂F ∂ F fp = (A + ∂) F = 1 − + 2 ... . (A.3) ∆E ≈ Ω∆L ≈ Ω~ and thus suffer phase shifts differing A A A by about ∆ϕ ≈ ∆Et/~ ≈ Ωt, where t is the time of flight. For a typical t = 1s this implies a phase shift of Operating on (A.3) with (A + ∂) obviously gives F . order 10−13rad, which is orders of magnitude less than To further justify the above formal operations we may solve (A.1) in a different way. An integrating factor is presently detectable [28]. Ax In addition to the small size of gravitomagnetic effects easily seen to be e , so one might see in the laboratory there is a serious fur- ∂(eAxf) = eAx(A + ∂)f = eAxF. (A.4) ther inherent difficulty in almost any such experiment; a rotation of the apparatus would in general have similar Integration then gives the general solution effects and swamp the gravitomagnetic effects, so such rotations would have to be controlled and compensated x Z 0 to very high accuracy as mentioned in the introduction f = e−Ax e−Ax F (x0)dx0 + Ce−Ax. (A.5) and in ref. [15]. The results of the GPB experiment and the theoreti- Since (A.1) is linear and F is assumed to be expandable cal results of this paper and ref.[15] are probably most in a power series we need only consider powers, F = xn. important in establishing the validity and consistency of Then we easily evaluate (A.5) using integration by parts, general relativity and the gravitomagnetic effects that it to obtain implies. Such gavitomagnetic effects are quite small in earth-based labs and satellite systems, as is clear from 1 xn nxn−1 n(n − 1)xn−2 f = − + ... + 1 + Ce−Ax. (8.3), but may play a large role in astrophysical phenom- A A A2 A3 ena such as the jets observed in active galactic nuclei, for (A.6) which the gravitomagnetic fields are much stronger [8]. This agrees with the power series for fp given in (A.3).

9. SUMMARY AND CONCLUSIONS Appendix B. ENERGY MOMENTUM TENSOR We have developed the theory of a spin 1/2 Dirac FOR THE DIRAC FIELD particle in a Riemann space and its weak field limit in considerable detail. In the low velocity limit for the par- We wish to obtain the energy momentum tensor for ticle the energies due to the Newtonian or gravitoelectric a Dirac field in flat space, which occurs in (6.8) and 11

(6.9)[16]. We begin with the Lagrangian (3.1) for the To verify the result (B.5) we may calculate the divergence free Dirac ﬁeld and work out the canonical energy mo- of T µν to ﬁnd, after some algebra, that it gives the correct mentum tensor according to the Noether theorem; it is, Lorentz force, up to a constant multiplier C,

µ ∂L ∂L ¯ µ T ν = C[ ψ|ν + ¯ ψ|ν − δν L] ∂ψ|µ ∂ψ|µ µν µα ¯ µα T |µ = −jαF = −(ψγαψ)F (B.6) ¯ µ ¯ µ = C[aψiγ ψ|ν − bψ|ν iγ ψ]. (B.1) where we have omitted the term proportional to L since it is zero for a solution of the free Dirac equation. Using In the interaction Lagrangian (6.8) the energy momen- the fact that the Dirac and the Klein-Gordon equations tum tensor is contracted with the symmetric h so the are obeyed by ψ we calculate the two divergences of this µν symmetrization in (B.5) is not relevant. tensor to be

µν µν 2 ¯ µ ¯|ν µ T |µ = 0,T |ν = C(b − a)[m (ψiγ ψ) − (ψ iγ ψ|ν )] (B.2)

If we choose b = a, as in the text, both divergences are zero and the tensor has symmetry in ψ and ψ¯. Moreover we may then consistently symmetrize T µν and have 1 T µν = [ψiγ¯ µψ|ν − ψ¯|ν iγµψ + ψiγ¯ ν ψ|µ − ψ¯|µiγν ψ] Acknowledgements 4 (B.3)

This has now been normalized so that in the low velocity This work was partially supported by NASA grant 8- limit 39225 to Gravity Probe B and by NSC of Taiwan under Project No. NSC 97-2112-M-002-026-MY3. Pisin Chen T 00 ≈ mψψ¯ (B.4) thanks Taiwan’s National Center for Theoretical Sciences for their support. Thanks go to Robert Wagoner, Francis Finally, to include the electromagnetic ﬁeld we use the Everitt, and Alex Silbergleit and other members of the minimal substitution recipe i∂µ → i∂µ − eAµ to get Gravity Probe B theory group for useful discussions, and to Mark Kasevich of the Stanford physics department for 1 T µν = [ψiγ¯ µψ|ν − ψ¯|ν iγµψ + ψiγ¯ ν ψ|µ − ψ¯|µiγν ψ] interesting comments on atomic beam interferometry and 4 equivalence principle experiments. Kung-Yi Su provided 1 − [eψA¯ ν γµψ + eψA¯ µγν ψ] (B.5) valuable help with the manuscript. Elisa Varani thanks 2 Cavallo Paciﬁc for encouragement and support.

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