aa r X i v : . [ phy s i c s . g e n - ph ] J u l Scattering Relativity in Quantum Mechanics
Richard Shurtleff ∗ October 11, 2018
Abstract
By adding generalizations involving translations, the machinery of the quantumtheory of free fields leads to the semiclassical equations of motion for a charged mas-sive particle in electromagnetic and gravitational fields. With the particle field trans-lated along one displacement, particle states are translated along a possibly differentdisplacement. Arbitrary phase results. And particle momentum, a spin (1/2,1/2)quantity, is allowed to change when field and states are translated. It is shown that apath of extreme phase obeys a semiclassical equation for force with derived terms thatcan describe electromagnetism and gravitation.Keywords: Special Relativity, Quantum Fields, Lorentz Force Law, GeodesicsPACS numbers: 03.70.+k, 03.65.Sq, 11.30.Cp,
Quantum fields differ from quantum states. While fields are sums of the creation and an-nihilation operators that add or remove states from multiparticle states, fields transform bynonunitary representations (reps) of spacetime symmetries while states and their operatorstransform with unitary reps.In a way, states and fields differ somewhat like identical experiments in different labs,say one in lab B and one in lab C. States and fields describe the same physical situationbut are somewhat isolated from one another. So, like experiments in labs B and C, there is ∗ affiliation and mailing address: Department of Sciences, Wentworth Institute of Technology, 550 Hunting-ton Avenue, Boston, MA, USA, ZIP 02115, telephone number: (617) 989-4338, fax number: (617) 989-4591, e-mail address: shurtleff[email protected] INTRODUCTION b and states and their operators by a possibly different amount b S . Arbitrary displacements bring arbitrariness to the phase, the scalar product of momentumand location p · x. The arbitrariness is shown to introduce general relativistic aspects to themotion such as local and general coordinate systems and Christoffel connections.The analogy with experiments in labs B and C is not pushed too far. One follows theanalogy to generalize a standard process of determining fields. Since these are generalizations,conventional results are the default.Another generalization involves reps of translations for various spins. In particular mo-mentum is a 4-vector, with spin (1/2,1/2), and can be changed under a translation by suchreps when the momentum is linked to a second rank tensor, with spins (0,0),(1,0), (0,1),and (1,1). The linked tensor is arbitrary, an array of free variables to be constrained by theassumptions. It turns out to be related to the electromagnetic field.The results of the generalized calculation are evaluated by rudimentary methods ap-propriate to the introduction of new ideas. Paths are deduced semiclassically by followingextreme phase.Fields describing massive particles are sums of coefficient functions times operators. InSec. 2, the generalizations are applied to a conventional process that derives expressions forthe coefficient functions. We follow Ref. [1] closely. Paths of extreme phase are obtainedin Sec. 3 from the expression for phase in Sec. 2. In Sec. 4, parallel translations of thespin (1/2,1/2) particle momentum are developed, making a momentum at an initial locationequivalent to a possibly different momentum at some other location. The parallel translationof momentum is path dependent. In Sec. 5, the arbitrariness in phase accompanying the dualdisplacements for states and fields is developed and is shown to describe curved spacetime.The path of extreme phase obeys a semiclassical force equation that is shown in Sec. 6 tocorrespond to the motion of a charged massive particle in a combined electromagnetic andgravitational field.
FIELD AND STATE This section supplies a derivation of some properties of the quantum field of a massiveparticle species, following Ref. [1] closely. But fields are allowed to be transformed by repsof translations. We do not agree that fields must be translation invariants.[2, 3]The quantum field ψ l ( x ) for a species of particles of mass m and spin j is constructed asa linear combination of annihilation and creation operators, ψ l ( x ) = κψ + l ( x ) + µψ − l ( x ) . Onehas an annihilation field ψ + and a creation field ψ − given by ψ + l ( x ) = Z d p u lσ ( x, −→ p ) a σ ( −→ p ) ,ψ − l ( x ) = Z d p v lσ ( x, −→ p ) a † σ ( −→ p ) , (1)where the repeated index σ is summed, a σ ( −→ p ) and a † σ ( −→ p ) are operators that remove or addan eigenstate of momentum −→ p and spin component σ. The spatial components −→ p are freewhile the time component of momentum, i.e. energy, is found from p t = q m − −→ p . The coefficient functions u and v are constrained by the ways quantities in (1) transformunder Poincar´e transformations to a new spacetime reference frame: ( i ) the operators a and a † transform with a unitary representation (rep), ( ii ) the coefficients u and v are required tobe invariant and ( iii ) the quantum field transforms by a nonunitary rep.We part company with Weinberg by scattering the spacetime transformations of fieldsand states, using the equivalence of inertial frames to free the frame for fields from the framefor states. The reference frame for the fields undergoes a Poincar´e transformation to a newframe, while the frame for the states and operators is transformed to their new frame.Let the reference frame of the fields be related to the frame for the states and operators bya Lorentz transformation λ. The new frame for the fields is obtained by applying a Lorentztransformation Λ , while the new state and operator frame is obtained with Λ S , where S indicates ‘States’. Call the coordinates x for the field frame and x S for the state frame. Wehave x = λx S and x ′ = Λ x = λx ′ S and Λ = λ Λ S λ − . (2)The similarity transformation λ relating Λ S and Λ means that they are equivalent transfor-mations, but in different inertial reference frames.That takes care of rotations and boosts, now for translations. When the fields are trans-lated along the displacement b to get to their new reference frame, the states are translatedalong some possibly different displacement b S . The description of an experiment depends on relative coordinates in which any globaldisplacement, like b or b S , cancels out. Thus b and b S can be completely arbitrary. Instead, FIELD AND STATE b S for the states depends on theLorentz transformation Λ , the event x, and the displacement of the field b.b S = b S (Λ , x, b ) . (3)Conventionally, the two displacements would agree, i.e. b S → b. Thus, for a Poincar´e transformation of the field ψ, the operators a and a † (and states)transform by (Λ S , b S ) = ( λ − Λ λ, b S (Λ , x, b )) . The unitary transformation U (Λ S , b S ) appliedto operators yields U (Λ S , b S ) ψ + l ( x ) U − (Λ S , b S ) = Z d p u lσ ( x, −→ p ) e i Λ S p · b S s (Λ S p ) t p t D ( j ) σ ¯ σ ( W − ) a ¯ σ ( −−→ Λ S p ) , (4)where the dot indicates the scalar product p · x ≡ η αβ p α x β with flat spacetime metric η, D ( j ) is a spin j unitary representation of rotations, W (Λ S , −→ p ) is the Wigner rotation for Λ S : k → p → Λ S p → k and k = (0 , , , m ) . The coefficients u are invariant.Contrary to the unitary transformation U (Λ S , b S ) of operators, the fields transform witha nonunitary rep D (Λ , b ) ,U (Λ S , b S ) ψ + l ( x ) U − (Λ S , b S ) = D − l ¯ l (Λ , b ) ψ +¯ l (Λ x + b ) , (5)where Λ = λ Λ S λ − by (2) with λ the state frame to field frame transformation and Λ x + b is the location in the new field frame of the event x. The expression for ψ − differs from the expression for ψ + only by v for u, − i for i, and D ( j ) ∗ for D ( j ) . It makes no sense here to write expressions for both; henceforth considermainly ψ + . The discussion is similar for ψ − , except for some special considerations with D ( j ) ∗ , see [1] for details.Not every Lorentz rep D (Λ ,
0) can be matched with any Poincar´e rep D (Λ , b ) , whichincludes a translation along b. There are some requirements.[4, 5] The nontrivial translationssought here require reducible Lorentz reps D (Λ , D (Λ ,
0) must be of the form (
A, B ) ⊕ ( C, D ) with spins (
A, B ) linked to (
C, D ) . Spins (
A, B ) and (
C, D ) are linked when(
C, D ) = ( A ± / , B ± / . (6)For example, the Dirac 4-spinor, spin (1 / , ⊕ (0 , / , has linked spins.Following the conventional process, the confluence of unitary and nonunitary transfor-mations of operators and fields in the sums (1) yields expressions for the coefficient functions u ( x, −→ p ) . The standard process is to write the fields on the left and right in (5) with the sums
FIELD AND STATE −→ p to −−→ Λ S p, and equate integrands. Onegets e i (Λ S p ) · b S s p t (Λ S p ) t D l ¯ l (Λ , b ) u ¯ lσ ( x, −→ p ) = u l ¯ σ (Λ x + b, −−→ Λ S p ) D ( j )¯ σσ ( W (Λ s , −→ p )) , (7)where the D ( j ) and D have changed sides to avoid the inverses seen in (4) and (5).Both u ¯ lσ ( x, −→ p ) and u l ¯ σ (Λ x + b, −−→ Λ S p ) can be written in terms of the coefficient functionsat the origin x = 0. In (7) substitute Λ = 1 and b = − x, so that Λ x + b = 0, Λ S = 1, and W = 1. We get u ¯ lσ ( x, −→ p ) = e − ip · b S (1 ,x, − x ) D ¯ ll (1 , + x ) u lσ (0 , −→ p ) . (8)Then put Λ = 1, with x = ˜Λ˜ x + ˜ b and b = − ˜Λ˜ x − ˜ b, so that again we have Λ x + b = 0, Λ S = 1, and W = 1. Drop the tildes. This gives u ¯ lσ (Λ x + b, −→ p ) = e − ip · b S (1 , Λ x + b, − Λ x − b ) D ¯ ll (1 , Λ x + b ) u lσ (0 , −→ p ) , or, by resetting the momentum p, p → Λ S p,u ¯ lσ (Λ x + b, −−→ Λ S p ) = e − i Λ S p · b S (1 , Λ x + b, − Λ x − b ) D ¯ ll (1 , Λ x + b ) u lσ (0 , −−→ Λ S p ) . (9)Substituting expressions (8) and (9) back in (7) and taking steps to put all x - and b -dependence on the left side gives e + i Λ S p · [ b S (Λ ,x,b ) − Λ S b S (1 ,x, − x )+ b S (1 , Λ x + b, − Λ x − b )] D l ¯ l (Λ , u ¯ lσ (0 , −→ p ) = s (Λ S p ) t p t u l ¯ σ (0 , −−→ Λ S p ) D ( j )¯ σσ ( W (Λ S , −→ p )) . (10)The exponent on the left shows the derivation history with Λ = 1 and b = − x in b S (1 , x, − x )and Λ = 1 with b = − x = − ˜Λ˜ x − ˜ b in b S (1 , Λ x + b, − Λ x − b ) . Since x and b are confined to the left in (10), the function b S (Λ , x, b ) makes the 4-vector V (Λ , x, b ) ≡ b S (Λ , x, b ) − Λ S b S (1 , x, − x ) + b S (1 , Λ x + b, − Λ x − b )independent of x and b. To do this, one can show that b S (Λ , x, b ) must be in the followingform, b µS (Λ , x, b ) = b µ (Λ) − Λ µS σ [ A ( x )] σν x ν + [ A (Λ x + b )] µν (Λ x + b ) ν , (11)where A ( x ) is an arbitrary second rank tensor field. By (11), we have V (Λ , x, b ) ≡ b (Λ) − Λ S b (1) + b (1) , PATHS OF EXTREME PHASE x and b, as was required. For simplicity, drop the constant, b µ = 0.One recovers b S = b when λ = 1 and A is the identity, A µν = δ µν , where δ is one for equalindices and zero otherwise. Also, note that b S depends on the values of the field A at both x and Λ x + b, the old and the new coordinates of the event.By the expression for b µS (Λ , x, b ) in (11), we have b αS (1 , x, − x ) = − A αµ x µ . Then u ¯ lσ ( x, −→ p )in (8) becomes u lσ ( x, −→ p ) = s mp t e ip · Ax D l ¯ l ( L, x ) u ¯ lσ (0 , −→ . (12)To get to the rest momentum −→ p = −→ , which is the 4-vector k = (0 , , , m ) , we considered(10) with p = k and with Λ S = L S = λ − Lλ is a transformation taking k to p so that W ( L S , −→ ψ + l ( x ) , (1) and the similar expression for v lσ ( x, −→ p )in for ψ − l ( x ) , determines many aspects of the quantum field ψ l ( x ) . The complications offurther developments along these lines[1] are not followed here. Here we investigate theeffects of the arbitrary field A on the phase. Paths with extreme phase are the most likely, since deviations from the path introducesinterference.[6] The most likely paths are expected to be the paths of particles in classicalmechanics. Successfully describing classical motion is a first step in quantum theory.Consider intervals δx so short that A varies negligibly along δx. Also assume that p iseffectively constant along δx. Then the change in the phase Θ associated with a particle ofmomentum p is, by (12), δ Θ = p · Aδx = η αβ p α A βµ δx µ (13)over such a displacement δx. Let ˆ p be the timelike unit vector p/m and let the spacelike unit vector ˆ p ⊥ have a null scalarproduct with ˆ p. Denote the magnitude of
Aδx by δτ. In detail, η αβ ˆ p α ˆ p β = − , η αβ ˆ p α ⊥ ˆ p β ⊥ =+1 , η αβ ( Aδx ) α ( Aδx ) β = − δτ . Thus,
Aδx = δτ (cosh φ ˆ p + sinh φ ˆ p ⊥ ) , and δ Θ = − mδτ cosh φ. The extreme value of cosh φ occurs for φ = 0, so the extreme value of δ Θ is given by δ Θ extreme = − mδτ, and one has A αµ δX µ = A αµ δx µ extreme = m − p α δτ , (extreme δ Θ) . (14)The upper case δX = δx extreme indicates a path of extreme phase Θ . The extreme path occurswhen A αµ δX µ is parallel to the momentum p. PARALLEL TRANSLATIONS X ( τ ) continue by attaching a second δX, perhapsfor slightly different A and p, to the δX just found. The process forms a curve X ( τ ) ofextreme phase with A αµ ( τ ) and p α ( τ ) the values of A and p along the curve X ( τ ) . Indicating the derivative with respect to τ with a dot,˙ X ≡ dX/dτ , (15)(14) becomes p α = mA αµ ˙ X µ . (Extreme phase) (16)This equation relates the particle momentum p to the velocity, the tangent ˙ X, along thepath for extreme phase.We can get ‘local coordinates’ ξ ( x ) if A is the transformation field A αµ = ∂ξ α ∂x µ . (Local coordinate transformation) (17)A neighborhood of the event x , x = x + δx, is mapped into a neighborhood of ξ ( x ) by ξ α ( x ) = ξ α ( x + δx ) = ξ α + ∂ξ α ∂x µ δx µ = ξ α + A αµ δx µ , where higher order terms in δx are dropped.Not all tensor fields allow such an interpretation. The second order partials must com-mute, ∂ ξ α /∂x λ ∂x µ = ∂ ξ α /∂x µ ∂x λ , or ∂A αµ ∂x λ = ∂A αλ ∂x µ . (Integrability conditions) (18)These are integrability conditions when, as is assumed, the field A is a field of transforma-tions.By (16) and (17), the curve X ( τ ) transforms to the curve Ξ α ( τ ) = ξ α ( X ( τ )) of extremephase in coordinates ξ with p α = m ˙Ξ α = mA αµ ˙ X µ , (19)where uppercase Ξ( τ ) denotes the path of extreme phase in local coordinates ξ. Next the concept of parallel translation of the momentum is introduced to determine howthe momentum changes with location. Changing momentum with location alters the path X ( τ ) of extreme phase.Momentum is a 4-vector. Rotations and boosts mix the components of the 4-vectorby presumably familiar rotation and boost matrices, a spin (1 / , /
2) representation of theLorentz group. See, for example Ref. [7] Chap. 10.
PARALLEL TRANSLATIONS
C, D ) to be linked to vector spin (
A, B ) = (1 / , /
2) if C = A ± / D = B ± / . Thus (
C, D ) ∈ { (0 , , (1 , , (0 , , (1 , } , which is the spin composition of a second ranktensor. Thus translation links the momentum to some second rank tensor T together makinga 4 + 16 = 20-component quantity Φ , Φ = (cid:18) p α T γδ (Φ) (cid:19) . (20)Like A , T (Φ) is free, a collection of 16 free parameters, to be constrained as needed.To generate the rotations and boosts of Φ , select a standard rep of the Lorentz groupwith angular momentum and boost generators. The choice here is from Ref. [8], J ρσ = − i (cid:18) η σµ δ ρν − η ρµ δ σν
00 + η ργ δ σǫ δ δξ − η σγ δ ρǫ δ δξ + η ρδ δ σξ δ γǫ − η σδ δ ρξ δ γǫ (cid:19) . (21)Then translations are generated with momentum matrices P µ , [9] P σ = (cid:18) P σ (cid:19) = i (cid:18) π δ σγ δ αδ + π δ σδ δ αγ + π η σα η γδ + π η σρ η ακ ǫ ρκγδ (cid:19) , (22)where ǫ is the antisymmetric symbol and there are four constants π i because the transforma-tion of a second rank tensor combines the four irreducible reps { (0 , , (1 , , (0 , , (1 , } . There is another set of matrices that change T (Φ) and leave p unchanged. But we don’twant that. The momentum matrices, displayed above with only the 12-block nonzero, changethe four-vector momentum p α and leave the tensor T (Φ) unchanged. Keeping both blocks,so both p α and T (Φ) change, makes the momentum matrices no longer commute, [ P µ , P ν ] =0, and translations would not commute, which is deemed unacceptable spacetime behaviorand violates the Poincar´e algebra.[1, 7] The needed momentum matrices are those in (22).Since the momentum matrices P µ have nonzero components only in an off-diagonal block,the product of any two vanishes, P µ P ν = 0, and any function of the P µ s that can be expandedin a power series reduces to a linear function. Thus the translation matrix for a translationalong a displacement δx is Q ( δx ) = exp ( − iδx σ P σ ) = − iδx σ P σ , where is the 20 ×
20 unit matrix. The translation of a four-vector v α yields v ′ α , with v ′ α = [ Q ( δx )] ασ v σ = v α − iη σµ ( P σ ) αβγ T βγ (Φ) δx µ = v α + η σµ T ασ δx µ , (23) PARALLEL TRANSLATIONS T is an abbreviation, T ασ ≡ − i ( P σ ) αβγ T βγ (Φ) . (24)Translation adds an inhomogeneous term T δx to v. The added term is the same for any4-vector v even if the components of v vanish. Also, translations are path dependent. Alarge displacement is the result of various sequences of small displacements, so over finiteintervals the translated four-vector v ′ may depend on path.The question arises: What four-vector at x + δx is equivalent to the four-vector at x ? Isit the four-vector with the same components or the translated four-vector? We assume it isthe translated four-vector, so that simple translation does not produce any innate change tothe four-vector. Parallel Translation of a Four-Vector. The translated four-vector v ′ is equivalent to theoriginal four-vector v. The particle momentum is a 4-vector. The momentum at a nearby event that is equivalentto the momentum at an original event should be obtained by parallel translation.
Dynamical Postulate. A particle in a given eigenstate of momentum p remains in eigen-states of equivalent momenta as spacetime is translated. The coefficient functions u lσ ( x, −→ p ) in (12) are referenced to the coefficient function at an‘origin’ x = 0. Suppose the origin is on the semiclassical path of extreme phase X ( τ ) . As theparticle moves along X ( τ ) , its momenta change to equivalent momenta. The interval δX in(14) for the extreme phase change δ Θ = p · AδX is followed by the interval δX ′ obtainedfor the extreme phase change δ Θ ′ = p ′ · AδX ′ with the momentum p ′ obtained by paralleltranslation along δX, a special case of (23).Therefore, the translated momentum p ′ is p ′ α = p α + η σµ T ασ δX µ . (25)Since the path X ( τ ) is a succession of intervals and the momentum is parallel translatedalong each interval, the momentum is a function p ( τ ) of proper time τ, whose derivative withrespect to τ is determined by parallel translation (25) to be˙ p α = η σµ T ασ ˙ X µ . (Parallel translation) (26)The tensor T is a free parameter that is constrained by the other equations satisfied by themomentum p and the path of extreme phase X. Equation (26) is the semi-classical equationof motion.
CURVED SPACETIME Curved spacetime is a consequence of extreme phase and the fact that mass is the magnitudeof the momentum. Substitute the requirement of extreme phase p = mA ˙ X , by (16), intothe mass equation, η αβ p α p β = − m , (27)1 m η αβ p α p β = η αβ A αµ A βν ˙ X µ ˙ X ν = − . By collecting some of the quantities together in g µν , one has g µν ˙ X µ ˙ X ν = − , (28)where the tensor field g µν ( x ) and its inverse are defined in terms of the tensor field A ( x ) by g µν ≡ η αβ A αµ A βν and g µν ≡ η αβ A − µα A − νβ , (29)with g µσ g σρ = δ ρµ , as is easily verified. Since the flat spacetime metric η αβ is symmetric, both g µν and g ρσ are symmetric. We call g µν the ‘curved spacetime metric’.By (28), (29), and ˙Ξ α = A αµ ˙ X µ from (19), one sees that η αβ ˙Ξ α ˙Ξ β = − , (30)where Ξ( τ ) is the path of extreme phase in coordinates ξ. Because the metric in (30) is the flat spacetime metric η, the transformation A giveslocally flat spacetime coordinates ξ from curved spacetime coordinates x. It follows that g µν is locally Lorentzian.Having two metrics makes raising and lowering indices a possible source of confusion.Thus raising and lowering indices is kept to a minimum and, when needed, the metricinvolved is displayed clearly.Furthermore, we can transform the path Ξ( τ ) at the event O at τ = 0 to a frame so that˙Ξ(0) has only its time component nonzero, ˙Ξ(0) = (0 , , , ˙Ξ t ) . Then, by (30), we have ˙Ξ t = 1 and d Ξ t = dτ, so we are again justified in calling the quantity τ the ‘proper time’ alongthe path X ( τ ) of extreme phase. The parameter τ is the time in a local Lorentz frame inwhich the particle is momentarily at rest.Turn now to the particle momentum. Define a 4-vector ¯ p µ ( τ ) by¯ p µ ≡ m ˙ X µ . (31)By (28), one finds that g µν ¯ p µ ¯ p ν = − m . (32) COMBINED ELECTROMAGNETIC/GRAVITATIONAL MOTION m is the magnitude of the momentum ¯ p calculated with the metric g and wesee that ¯ p µ ( τ ) is the ‘curved spacetime momentum’ of the particle along the path X ( τ ) ofextreme phase. And p α is the flat spacetime particle momentum because of the flat spacetimemetric η in (27), η αβ p α p β = − m . By (16) and (31), it follows that p α = A αµ ¯ p µ . (33)Thus the flat and curved spacetime momenta are related by the same transformation A that,by (19), relates the tangents of the flat spacetime and curved spacetime paths ˙Ξ and ˙ X. In this section, the semi-classical equation of motion (26) is shown to describe the motion ofa charged particle in gravitational and electromagnetic fields.The path of extreme phase X µ is the ‘curved’ spacetime path because of the curvedspacetime metric g in g µν ˙ X µ ˙ X ν = − , (28). And the momentum p α is the flat spacetimeparticle momentum because of the flat spacetime metric η in the mass equation (27), η αβ p α p β = − m . Thus the equation of motion (26), ˙ p α = η σµ T ασ ˙ X µ , mixes the flat spacetime particlemomentum p α and the curved spacetime path X µ . In (33), p α is expressed in terms of the curved spacetime momentum, p α = A αµ ¯ p µ , and bysubstituting in the equation of motion (26), we can arrange to have an equation of motionwith the curved spacetime quantities ¯ p µ and X µ . One finds that¨ X µ = 1 m ˙¯ p µ = 1 m A − µα η σν T ασ ˙ X ν − A − µβ ∂A βν ∂x λ ˙ X λ ˙ X ν , (34)where we used (16), p α = mA αµ ˙ X µ , to write p in terms of ˙ X. The term quadratic in velocity, ˙ X λ ˙ X µ , looks a bit like the acceleration in a gravitationalfield. Write A − µβ ∂A βν ∂x λ ˙ X λ ˙ X ν = 12 A − νβ ∂A βν ∂x λ + ∂A βλ ∂x ν ! ˙ X λ ˙ X ν = C µλν ˙ X λ ˙ X µ , (35)where C is given by C µλν = 12 A − µβ ∂A βν ∂x λ + ∂A βλ ∂x ν ! . (36)Clearly, the quantity C µλν is symmetric in its lower indices, C µλν = C µνλ . COMBINED ELECTROMAGNETIC/GRAVITATIONAL MOTION C is indeed the Christoffel connection of the metric g. The Christoffelsymbol of the first kind is defined to be [10][ µν, ρ ] ≡ ∂g µρ ∂x ν + ∂g νρ ∂x µ − ∂g µν ∂x ρ ! . By the definitions of C µλν in (36) and g µν in (29), i.e. g µν = η αβ A αµ A βν , one can show that[ µν, ρ ] = g ρσ C σµν + 12 η αβ " A αµ ∂A βρ ∂x ν − ∂A βν ∂x ρ ! + A αν ∂A βρ ∂x µ − ∂A βµ ∂x ρ ! . Looking at this, we see that C is the Christoffel connection when the terms in parenthesesvanish, i.e. when ∂A βρ ∂x ν = ∂A βν ∂x ρ . But this is just the integrability condition (18) so that A βν = ∂ξ β /∂x ν with A βν transformingcoordinates x to ξ β ( x ) as in (17).Thus, when A is a field of transformations to local coordinates, the quantity C is theChristoffel connection of the metric g, [10] C σµν = g ρσ [ µν, ρ ] = g ρσ ∂g µρ ∂x ν + ∂g νρ ∂x µ − ∂g µν ∂x ρ ! . (37)The ‘covariant derivative’ of ˙ X with respect to τ is defined to be[8, 11] D ˙ X µ dτ ≡ ¨ X µ + C µλν ˙ X λ ˙ X ν . (38)With this, the equation of motion (34) with (35) can be written as D ˙ X µ dτ = 1 m A − µα η σν T ασ ˙ X ν . (39)The covariant derivative on the left has the same form in any coordinate system.If the arbitrary tensor field T is chosen properly, one can have the term on the right in(39) be the electromagnetic Lorentz force divided by mass m . According to Ref. [12], weneed T to satisfy A − µα η σν T ασ ˙ X ν = qg σν F µσ ˙ X ν (40)for a particle of charge q in an electromagnetic field F. Thus the tensor field T in (24) shouldbe η σν T ασ = qA αλ g ρν F λρ . (41) COMBINED ELECTROMAGNETIC/GRAVITATIONAL MOTION b and b S for fields and states are equal, whichis the conventional assumption. By (11) and (29), that happens when A αλ = δ αλ and g ρν = η ρν , so that T simplifies to T ασ = qF ασ . Now the equation of motion (34) is D ˙ X µ dτ = qm g σν F µσ ˙ X ν , (42)which is the Lorentz force equation in curved spacetime.[12] This equation is covariant; ithas the same same form when the metric g is transformed to some other metric g ′ . [13]Since (39) is covariant, it can be transformed from x -coordinates to local ξ -coordinatesvia the transformation field A αν = ∂ξ α /∂x ν . One finds that¨Ξ α = qm η βκ f αβ ˙Ξ κ , (43)where ¨Ξ α = d ˙Ξ α /dτ, ˙Ξ α = A αν ˙ X ν , f αβ = A αν A βσ F νσ , and we used g σν = η αβ A αν A βσ by (29).While the fields f and F have been called the electromagnetic field in local and generalcoordinates, it has not yet been shown that they must be antisymmetric. This will beundertaken now.Take the derivative of η ˙Ξ ˙Ξ = − , (30), with respect to proper time τ ,0 = 2 η αβ ˙Ξ α ¨Ξ β = 2 m η αβ ˙Ξ α qη σµ f βσ ˙Ξ µ , so that, after multiplying by m/q, (cid:16) η αβ ˙Ξ α (cid:17) (cid:16) η µσ ˙Ξ µ (cid:17) f βσ = ˙Ξ β ˙Ξ σ (cid:16) f βσ + f σβ (cid:17) , where ˙Ξ β = η αβ ˙Ξ α . Thus f is antisymmetric along the path of extreme phase. Since f αβ = A αν A βσ F νσ , if f is antisymmetric, then F is antisymmetric as well, f βσ = − f σβ and F µν = − F νµ . (44)The antisymmetry of f and F adds evidence to their identification as the electromagneticfield in local and general coordinates. The remaining problem of relating electromagneticand gravitational fields to sources may be treated elsewhere. EFERENCES References [1] Weinberg, S.,
The Quantum Theory of Fields , Vol. I (Cambridge University Press, Cam-bridge, 1995), Chapter 5.[2] See, for example, Raymond, P.,
Field Theory , (Benjamin/Cummings, London, 1981),p. 19 line 17.[3] Implicitly in S. Weinberg,
The Quantum Theory of Fields, Vol. I (Cambridge UniversityPress, Cambridge, 1995): Chapter 5, p. 192, (5.1.6,7), (Λ , a ) together on the left, (Λ − )alone on the right.[4] G. Ya. Lyubarskii, The Application of Group Theory in Physics (Pergamon Press, Oxford,1960): p. 310 after (74,4).[5] R. Shurtleff,
A Derivation of Vector and Momentum Matrices , online article inarXiv:math-ph/0401002[v3], 2 Jul 2007.[6] See, for example, Feynman, R.P., Hibbs, A.R.,
Quantum Mechanics and Path Integrals (McGraw-Hill, New York, N.Y., 1965): Sec. 2-3, p. 29.[7] W. Tung,
Group Theory in Physics (World Scientific Publishing Co. Pte. Ltd., Singapore,1985): p. 185 (10.2-16,17,18).[8] See, for example, Weinberg, S.,
Gravitation and Cosmology , (John Wiley & Sons, NewYork, 1972), Chap. 2, Sec. 12, p. 59.[9] R. Shurtleff,
Poincare Connections in Flat Spacetime , online article inarXiv:gr-qc/0502021v2, September 29, 2013.[10] See, for example, Adler, R., Bazin, M., Schiffer, M.,
An Introduction to General Rela-tivity , (McGraw-Hill, New York, 1965).[11] See, for example, Dirac, P. A. M.,
General Theory of Relativity , (Princeton UniversityPress, 1996), (originally published: J. Wiley, New York, 1975), Chapters 7 - 10.[12] See, for example, Weinberg, S.,
Gravitation and Cosmology , (John Wiley & Sons, NewYork, 1972), Eq. (5.1.11), p. 123 and Eq. (5.2.9,10), p. 125.[13] See, for example, Misner, C. W., Thorne, K. S., Wheeler, J. A.,