ii. covariant approach - pennsylvania state university

arX

iv:q

uant

-ph/

0107

002v

1 3

0 Ju

n 20

01LANL arXiv server, E-print No. quant-ph/0107002

Fibre bundle formulation ofrelativistic quantum mechanics

II. Covariant approach

Bozhidar Z. Iliev∗ † ‡

Short title: Bundle relativistic quantum mechanics: II

Basic ideas : → November 1997, January 1998

Began/Ended: → February 16, 1998/March 27, 1998

Initial typeset : → April 1–14, 1998

Revised: → August 1999

Last update : → May 13, 2001

Composing/Extracting part II : → April 22/April 24, 1998

Updating part II : → October, 1998, August 1999

Produced: → February 29, 2008

LANL arXiv server E-print No. : quant-ph/0107002

BO/•• HOr© TM

Subject Classes:

Relativistic quantum mechanics, Differential geometry

2000 MSC numbers:

81Q99, 81S992001 PACS numbers:

02.40.Ma, 02.40.Yy02.90.+p, 03.65.Pm

Key-Words:

Relativistic quantum mechanics, Fibre bundles,Geometrization of relativistic quantum mechanics,

Relativistic wave equations, Dirac equation, Klein-Gordon equation

∗Department Mathematical Modeling, Institute for Nuclear Research andNuclear Energy, Bulgarian Academy of Sciences, Boul. Tzarigradsko chaussee 72,1784 Sofia, Bulgaria

†E-mail address: [email protected]‡URL: http://theo.inrne.bas.bg/∼bozho/

http://arXiv.org/abs/quant-ph/0107002v1

Bozhidar Z. Iliev: Bundle relativistic quantum mechanics. II 1

Contents

1 Introduction 1

2 Dirac equation 2

3 Other relativistic wave equations 8

4 Propagators and evolution transports or operators 94.1 Green functions (review) . . . . . . . . . . . . . . . . . . . . . 94.2 Nonrelativistic case (Schrodinger equation) . . . . . . . . . . 114.3 Dirac equation . . . . . . . . . . . . . . . . . . . . . . . . . . 134.4 Klein-Gordon equation . . . . . . . . . . . . . . . . . . . . . . 14

5 Conclusion 16

Appendix A Linear transports along maps in fibre bundles 16

Appendix B Matrix operators 19

References 21This article ends at page . . . . . . . . . . . . . . . . . . . . . 22

Abstract

We propose a new fibre bundle formulation of the mathematical base ofrelativistic quantum mechanics. At the present stage the bundle form ofthe theory is equivalent to its conventional one, but it admits new types ofgeneralizations in different directions.

In the present second part of our investigation, we consider the covari-ant approach to bundle description of relativistic quantum mechanics. Init the wavefunctions are replaced with (state) sections of a suitably chosenvector bundle over space-time whose (standard) fibre is the space of thewavefunctions. Now the quantum evolution is described as a linear trans-portation (by means of the evolution transport along the identity map of thespace-time) of the state sections in the bundle (total) space. The equationsof these transportations turn to be the bundle versions of the correspond-ing relativistic wave equations. Connections between the (retarded) Greenfunctions of these equations and the evolution operators and transports arefound. Especially the Dirac and Klein-Gordon equations are considered.


1. Introduction

This paper is a second part of our investigation devoted to the fibre bun-dle description of relativistic quantum mechanics. It is a straightforwardcontinuation of [1].

The developed in [1] bundle formalism for relativistic quantum waveequations has the deficiency that it is not explicitly covariant; so it is notin harmony with the relativistic theory it represents. This is a consequenceof the direct applications of the bundle methods developed for the nonrel-ativistic region, where they work well enough, to the relativistic one. Thepresent paper is intended to mend this ‘defect’. Here we develop an appro-priate covariant bundle description of relativistic quantum mechanics whichcorresponds to the natural character of this theory. The difference betweenthe time-dependent and covariant approaches is approximately the same asthe one between the Hamiltonian and Lagrangian derivation (and forms) ofsome relativistic wave equation.

The organization of the material is the following.Sect. 2 contains a detailed covariant application of the ideas of the bun-

dle description of nonrelativistic quantum mechanics (see, e.g. [1, sect. 2]or [2, 3]) to Dirac equation. The bundle, where Dirac particles ‘live’, is avector bundle over space-time with the space of 4-spinors as a fibre; so herewe again work with the 4-spinor bundle of [1, sect. 4], but now the evolutionof a Dirac particle is described via the (Dirac) evolution transport whichis a linear transport along the identity map of space-time. The state of aDirac particle is represented by a section (not along paths!) of the 4-spinorbundle and is (linearly) transported by means of Dirac evolution transportin this bundle. The Dirac equation itself is transformed into a system of fourequations uniquely describing this transportation. We write different equiv-alent (bundle or not) covariant versions of Dirac equation. In particular,we have found an interesting algebraic form of the Dirac equation connect-ing the coefficients of Dirac evolution transport and the Dirac analogue ofthe Hamiltonian corresponding to a covariant Schrodinger-like form of Diracequation. This analogue plays a role of a gauge field and from it the wholetheory can be recovered.

In Sect. 3 we apply the covariant bundle approach to Klein-Gordon equa-tion. For this purpose we present a 5-dimensional representation of thisequation as a first-order Dirac-like equation to which mutatis mutandis thetheory of Sect. 2 can be transferred practically without changes.

The goal of Sect. 4 is to be revealed some connections between the re-tarded Green functions (≡propagators) of the relativistic wave equations andthe corresponding to them evolution operators and transports. Generallyspeaking, the evolution operators (resp. transports) admit representationas integral operators, the kernel of which is connected in a simple man-ner with the retarded Green function (resp. Green morphism of a bundle).


Subsect. 4.1 contains a brief general consideration of the Green functionsand their connection with the evolution transports, if any. In Subsect. 4.2,4.3, and 4.4 we derive the relations mentions for Schrodinger, Dirac, andKlein-Gordon equations, respectively.

Sect. 5 closes the paper with a brief summary of the main ideas under-lying the bundle description of relativistic quantum mechanics.

Appendix A contains some mathematical results concerning the theoryof (linear) transports along maps required for the present investigation.

In Appendix B are given certain formulae concerning matrix operators,i.e. matrices whose elements are operators, which naturally arise in relativis-tic quantum mechanics.

The notation of the present work is the the same as the one in [1] andwe are not going to recall it here.

The references to sections, equations, footnotes etc. from [1] are obtainedfrom their sequential numbers in [1] by adding in front of them the Romanone (I) and a dot as a separator. For instance, Sect. I.4 and (I.5.2) meanrespectively section 4 and equation (5.2) (equation 2 in Sect. 5) of [1].

Below, for reference purposes, we present a list of some essential equa-tions of [1] which are used in this paper. Following the just given convention,we retain their original reference numbers.

ψ(t) = U(t, t0)ψ(t0), (I.2.2)

Ψγ : t→ Ψγ(t) = l−1γ(t)

(ψ(t)

), Hγ : t→ Hγ(t) = l−1

γ(t) ◦ H ◦ lγ(t), (I.2.3)

Uγ(t, s) = l−1γ(t) ◦ U(t, s) ◦ lγ(s) : Fγ(s) → Fγ(t), s, t ∈ J, (I.2.4)

Γγ(t) :=[Γb

a(t; γ)]

= −1

i~Hm

γ (t), (I.2.10)

Aγ(t) = l−1γ(t) ◦ A(t) ◦ lγ(t) : Fγ(t) → Fγ(t). (I.2.11)

2. Dirac equation

The covariant Dirac equation [4, sect. 2.1.2], [5, chapter XX, § 8] for a (spin12) particle with mass m and electric charge e in an external electromagneticfield with 4-potential Aµ is

(i~D/ −mc114)ψ = 0, D/ := γµDµ, Dµ := ∂µ −e

i~cAµ. (2.1)

Here i ∈ C is the imaginary unit, ~ is the Plank constant (divided by 2π),114 = diag(1, 1, 1, 1) is the 4 × 4 unit matrix, ψ := (ψ0, ψ1, ψ2, ψ3) is (thematrix of the components of) a 4-spinor, γµ, µ = 0, 1, 2, 3, are the well knownDirac γ-matrices [4–6], and c is the velocity of light in vacuum. Since it isa first order partial differential equation, it admits evolution operator Uconnecting the values of its solutions at different spacetime points. More


precisely, if x1, x2 ∈ M , M being the spacetime, i.e. the Minkowski spaceM4, then

ψ(x2) = U(x2, x1)ψ(x1). (2.2)

Here U(x2, x1) is a 4 × 4 matrix operator (see Appendix B) defined as theunique solution of the initial-value problem1

(i~D/x −mc114)U(x, x0) = 0, U(x0, x0) = idF , x, x0 ∈M (2.3)

where D/ = ∂/ − ei~c

A/ with ∂/ := γµ∂/∂xµ and A/ := γµAµ, F is the space of4-spinors, and idX is the identity map of a set X.

Suppose (F, π,M) is a vector bundle with (total) bundle space F , pro-jection π : F → M , fibre F , and isomorphic fibres Fx := π−1(x), x ∈ M .There exist linear isomorphisms lx : Fx → F which we assume to be diffeo-morphisms; so Fx = l−1

x (F) are 4-dimensional vector spaces.To a state vector (spinor) ψ we assign a C1 section2 Ψ of (F, π,M), i.e.

Ψ ∈ Sec1(F, π,M), by (cf. (I.2.3))

Ψ(x) := l−1x

(ψ(x)

)∈ Fx := π−1(x), x ∈M. (2.4)

Since in (F, π,M) the state of a Dirac particle is described by Ψ, we callit state section; resp. (F, π,M) is the 4-spinor bundle. The description ofDirac particle via Ψ will be called bundle description. If it is known, theconventional spinor description is achieved by the spinor

ψ(x) := lx(Ψ(x)

)∈ F . (2.5)

Now the analogue of (2.2) is

Ψ(x2) = U (x2, x1)Ψ(x1), x2, x1 ∈M. (2.6)

with (cf. (I.2.4))

U (y, x) = l−1y ◦ U(y, x) ◦ lx : Fx → Fy, x, y ∈M. (2.7)

Since U is a 4 × 4 matrix operator satisfying (2.3) and, as can easily beproved, the equality

U(x3, x1) = U(x3, x2) ◦ U(x2, x1), x1, x2, x3 ∈M, (2.8)

1Generally U is an integral matrix operator whose kernel for the free Dirac equation isgiven in [4, sect.2.5.1, equations (2.107) and (2.108)].

2In contrast to the time-dependent approach [1] and nonrelativistic case [7] now Ψ issimply a section, not section along paths [7]. Physically this corresponds to the fact thatquantum objects do not have world lines (trajectories) in a classical sense [8].


is valid, the map (2.7) is linear and:

U (x3, x1) = U (x3, x2) ◦ U (x2, x1), x1, x2, x3 ∈M, (2.9)

U (x, x) = idFx, x ∈M. (2.10)

Consequently, by definition A.1, the map U : (y, x) → U (y, x) = K idMx→y is

a linear transport along the identity map idM of M in the bundle (F, π,M).Alternatively, as it is mentioned in Appendix A, this means that Lγ : (t, s) →Lγ

s→t := U (γ(t), γ(s)), s, t ∈ J is a flat linear transport along γ : J →M in(F, π,M). We call U the (Dirac) evolution transport.

Equation (2.6) simply means that Ψ is U -transported (along idM ) sec-tion of (F, π,M) (cf. [9, definition 5.1]). Writing (A.10) for the evolutiontransport U and applying the result to a state section given by (2.4), weeasily can prove that (2.6) is equivalent to

DµΨ = 0, µ = 0, 1, 2, 3 (2.11)

where, for brevity, we have put Dµ := D idMxµ which is the µ-th partial (sec-

tion-)derivation along the identity map (of the spacetime) assigned to theDirac evolution transport. Since these equations are equivalent to the Diracequation (2.1), we call them bundle (system of) Dirac equations. The factthat now we have four equations, not a single one, reflects the flatness of Uwhich, in its turn, reflects (due to the Heisenberg uncertainty relations) thenon-existence of classical trajectories (world lines) for Dirac particles.

Now we shall introduce local bases and take a local view of the above-de-scribed.

Let {fµ(x)} be a basis in F and {eµ(x)} be a basis in Fx, x ∈ M . Thematrices corresponding to vectors and/or linear maps (operators) in thesefields of bases will be denoted by the same (kernel) symbol but in boldface,

for instance: ψ :=(ψ0, ψ1, ψ2, ψ3

)⊤and lx(y) :=

[(lx)µν(y)

]are defined,

respectively, by ψ(x) =: ψµ(x)fµ(x) and lx(eν(x)

)=:

(lx(y)

)µ

νfµ(y). We

put lx := lx(x); in fact this will be the only case when the matrix of lxwill be required as we want the ‘physics in Fx’ to correspond to that of Fat x ∈ M . A very convenient choice is to put eµ(x) = l−1

x

(fµ(x)

); so then

lx = 114 :=[δµν

]= diag(1, 1, 1, 1) is the 4 × 4 unit matrix.

The matrix U(y, x) of U(y, x) is defined by

ψ(y) =: U(y, x)ψ(x) (2.12)

and it is independent of ψ (see (2.2), (B.3), (B.4), and (B.5)). The matrixelements of U (y, x) are defined via U (y, x)

(eµ(x)

)=: U λ

µ(y, x)eλ(y) and,due to (2.7), we have

U (y, x) = l−1y · U(y, x) · lx (2.13)


which is generically, like U , a matrix operator (see Appendix B).According to (A.12) the coefficients of the evolution transport form four

matrix operators

µΓ(x) :=[

µΓλν(x)

]3

λ,ν=0:=

∂U (x, y)

∂yµ

∣∣∣∣y=x

= F−1(x)∂F (x)

∂xµ(2.14)

where U (x, y) = F−1(y)F (x) and F (x) is the matrix of a linear map (iso-morphism) defining U according to theorem A.1.

Applying (A.5) and (2.14), we find

∂U (y, x)

∂yµ= − µΓ(y) ⊙U (y, x),

∂U (y, x)

∂xµ= U (y, x) ⊙ µΓ(x), (2.15)

where ⊙ denotes the introduced by (B.2) multiplication of matrix operators.Therefore

∂/yU (y, x) = −Γ/(y)U (y, x), Γ/(x) := γµ · µΓ(x). (2.16)

Similarly to the nonrelativistic case [10], to any operator A : F → F weassign a bundle morphism A : F → F by

Ax := A|Fx := l−1x ◦ A ◦ lx. (2.17)

Defining

Gµ(x) := l−1x ◦ γµ ◦ lx, dµ := l−1

x ◦ ∂µ ◦ lx, ∂µ :=∂

∂xµ(2.18)

and using the matrices Gµ(x) and Eµ(x) given via (B.12), we get

Gµ(x) = l−1x Gµ(x)lx, dµ := 114∂µ + l−1

x

(∂µlx + Eµ(x)lx

). (2.19)

(Here and below, for the sake of shortness, we sometimes omit the argumentx.) The anticommutation relations

GµGν + GνGµ = 2ηµν idF ,

GµGν + GνGµ = GµGν + GνGµ = 2ηµν114

(= γµγν + γνγµ

),

(2.20)

where [ηµν ] = diag(1,−1,−1,−1) = [ηµν ] is the Minkowski metric tensor,can be verified by means of (2.18), (2.19), (B.10), and the well know analo-gous relation for the γ-matrices (see, e.g. [6, chapter 2, equation (2.5)]).

For brevity, if aµ : F → F are morphisms, sums like Gµ ◦ aµ will bedenoted by ‘backslashing’ the kernel letter, a\ := Gµ ◦ aµ (cf. the ‘slashed’notation a/ := γµaµ). Similarly, we put a\ := Gµ(x)aµ for aµ ∈ GL(4,C). Itis almost evident (see (2.18)) that the morphism corresponding to ∂/ := γµ∂µ

is d\ = Gµ(x) ◦ dµ:

d\ = l−1x ◦ ∂/ ◦ lx. (2.21)


Using (2.19), we observe that

d\ = ∂\ + Gµ(x)l−1x

(∂µlx + Eµ(x)lx

)= ∂\ + Gµ(x)l−1

x (∂µlx) + E\(x),

Eµ = l−1x Eµlx.

(2.22)

Now we are completely prepared for a more detailed exploration of Diracequation (2.1) from bundle view-point.

First of all, we rewrite (2.1) as

i~∂/ψ = Dψ, D := mc114 +e

cA/. (2.23)

We claim that this is the covariant Schrodinger-like form of Dirac equation;∂/ is the analogue of the time derivation d/dt and D corresponds to theHamiltonian H. We call D the Dirac function, or simply, Diracian of aparticle described by Dirac equation.

Substituting (2.5) into (2.23), acting on the result from the left by l−1x ,

and using (2.18), we find the bundle form of (2.23) as

i~d/Ψ = DΨ (2.24)

with D ∈ Mor(F, π,M) being the (Dirac) bundle morphism assigned to theDiracian. We call it bundle Diracian. According to (2.17) it is defined by

Dx := D|Fx = l−1x ◦ D ◦ lx = mc idFx +

e

cA\x, (2.25)

where Aµ = Aµ idF ∈ Mor(F, π,M), Aµ|x = l−1x ◦ (Aµ|x idFx) ◦ lx = Aµ|x idFx

are the components of the bundle electromagnetic potential and

A\x = Gµ(x) ◦ Aµ|x = Gµ(x) ◦ (l−1x ◦ Aµ idFx ◦ lx)

= l−1x ◦ (A/|x idFx) ◦ lx = A\x. (2.26)

From (2.24), we get the straightforward bundle analogue of (2.23):3

i~∂\Ψ = DmΨ (2.27)

where (see (2.25))

Dm = D − i~(d\ − ∂\) = mc114 +e

cA\ − i~(d\ − ∂\) (2.28)

3We can not write (2.27) as i~∂\Ψ = · · · because the l.h.s. of such an equality willcontain derivatives like ∂µΨ(x) ≡ ∂Ψ(x)/∂xµ which are not (well) defined at all due toΨ(x) ∈ Fx 6= Fy ∋ Ψ(y) for x, y ∈M and x 6= y.


is the matrix-bundle Diracian4 whose explicit form, due to (2.22), is

Dm∣∣Fx

= mc114 +e

cA\|x − i~

(Gµ(x)l−1

x (∂µlx) + E\), Eµ = l−1

x Eµ(x)lx.

(2.29)

The matrix-bundle Diracian is closely connected with the matrices (2.14)formed from the coefficients of the Dirac evolution transport. Applying theformalism developed, we can establish this connection by two independentways, leading, of course, to one and the same final result.

The first way is to insert (2.6) into (2.24) and acting on the result fromthe left by l−1

x , we get the bundle version of the initial-value problem (2.3)(see (2.24) and (2.10)):

i~d\ ◦ U (x, x0) = D ◦ U (x, x0), U (x0, x0) = idFx0, (2.30)

or

i~∂\ ⊙U (x, x0) = Dm ⊙U (x, x0), U (x0, x0) = 114. (2.30′)

Since (2.15) implies (cf. (2.16))

∂\yU (y, x) = −Γ\(y) ⊙U (y, x), (2.31)

Γ\(x) := Gµ(x) · µΓ(x), (2.32)

from (2.30′) we conclude

Γ\(x) = −1

i~Dm(x)

(= −

1

i~(mc114 +

e

cA\) + Gµl−1

x (∂lx) + E\). (2.33)

So, the ‘backslashed’ matrix of the coefficients of the Dirac evolution trans-port coincides up to a constant with the matrix-bundle Diracian.5

The another way mentioned for obtaining (2.33) is to write (2.11) into amatrix form, which by virtue of (A.11) is

∂µΨ + µΓ · Ψ = 0, (2.34)

then multiplying from the left by Gµ(x), summing over µ, and comparingthe result with (2.27), we get (2.33).

From the first derivation of (2.33) one can easily see that, in fact, theequation (2.33) is equivalent to (2.30) and, consequently, to the initial Diracequation (2.1). Hence (2.33) is an algebraic bundle form of the Dirac equa-tion.

4The matrix operator Dm is the relativistic analogue of the matrix-bundle Hamilto-nian [10, sect. 2].

5Cf. an analogous result connecting the matrix of the coefficients of the evolution trans-port and the matrix-bundle Hamiltonian in bundle nonrelativistic quantum mechanics(see [10, equation (2.21] or (I.2.10)).


Here the following natural problem arises. As we have just proved, (2.33)is equivalent to the Dirac equation (2.1). So, if Γ\ is given (by (2.33)), wecan recover the transport U , but it is known [11–13] that U can uniquelybe reconstructed by the matrices (2.14). Hence the four matrices (2.14)have to be equivalent to the one matrix (2.32). Is this possible? The an-swer is positive.6 Actually, substituting (2.14) into (2.32) and taking intoaccount (2.33), we observe that7

i~∂\F−1 = DmF−1 (= −i~Γ\F−1). (2.35)

So, given Γ\ and taking F to be any solution of the matrix Dirac equa-tion (2.35) (with respect to F−1), we can calculate µΓ via (2.14). On the

opposite, given µΓ or F , then Γ\ := GµµΓ = −

(∂\F−1

)F .

3. Other relativistic wave equations

The relativistic-covariant Klein-Gordon equation for a (spinless) particle ofmass m and electric charge e in a presence of (external) electromagneticfield with 4-potential Aµ is [5, chapter XX, § 5, equation (30′)]

(DµDµ +

m2c2

~2

)φ = 0, Dµ = ηµνD

ν := ∂µ −e

i~cAµ. (3.1)

Since this is a second-order partial differential equation, it does not directlyadmit an evolution operator and adequate bundle formulation and interpre-tation. To obtain such a formulation, we have to rewrite (3.1) as a first-order(system of) partial differential equation(s) (cf. Sect. I.5).

Perhaps the best way to do this is to replace φ with a 5 × 1 matrixϕ = (ϕ0, . . . , ϕ4)⊤ and to introduce 5 × 5 Γ-matrices Γµ, µ = 0, 1, 2, 3 with

components(Γµ

)i

j, i, j = 0, 1, 2, 3, 4 such that (cf. [14, chapter I, equa-

tions (4.38) and (4.37)]):

ϕ = a

i~D0φi~D1φi~D2φi~D3φmcφ

,

(Γµ

)i

j=

1 for (i, j) = (µ, 4)

ηµµ for (i, j) = (4, µ)

0 otherwise

(3.2)

where the complex constant a 6= 0 is insignificant for us and can be (par-tially) fixed by an appropriate normalization of ϕ.

6The real reason for that fact is the flatness of the Dirac transport U , i.e., for instance,the fact that U (y, x) depends only on the initial and final points x and y, respectively,and not on other objects as some path (curve) connecting them. All this reflects thenonexistence of (classical) trajectories of quantum particles.

7In fact (2.35) is the integrability condition for (2.14) considered as a first-order systemof partial differential equations with respect to F (x).


Then a simple checking shows that (3.1) is equivalent to (cf. (2.1))

(i~ΓµDµ −mc115)ϕ = 0 (3.3)

with 115 = diag(1, 1, 1, 1, 1) being the 5 × 5 unit matrix, or (cf. (2.23))

i~Γµ∂µϕ = Kϕ, K := mc115 +e

cΓµAµ. (3.4)

Now it is evident that mutatis mutandis, taking Γµ for γµ, ϕ for ψ,etc., all the (bundle) machinery developed in Sect. 2 for the Dirac equationcan be applied to the Klein-Gordon equation in the form (3.3) practicallywithout changes. Since the transferring of the results obtained in Sect. 2 forDirac equation to Klein-Gordon one is absolutely trivial,8 we are not goingto present here the bundle description of the latter equation.

Since the relativistic wave equations for particles with spin greater than1/2 are versions or combinations of Dirac and Klein-Gordon equations [5,6,14,15], for them is mutatis mutandis applicable the bundle approach devel-oped in Sect. 2 for Dirac equation or/and its version for Klein-Gordon onepointed above.

4. Propagators andevolution transports or operators

The propagators, called also propagator functions or Green functions, aresolutions of the wave equations with point-like unit source and satisfy ap-propriate (homogeneous) boundary conditions corresponding to the concreteproblem under exploration [4, 6]. Undoubtedly these functions play an im-portant role in the mathematical apparatus of (relativistic) quantum me-chanics and its physical interpretation [4, 6]. By this reason it is essentialto be investigated the connection between propagators and evolution opera-tors or/and transports. As we shall see below, the latter can be representedas integral operators whose kernel is connected in a simple way with thecorresponding propagator.

4.1. Green functions (review)

Generally [16, article “Green function”] the Green function g(x′, x) of alinear differential operator L (or of the equation Lu(x) = f(x)) is the kernelof the integral operator inverse to L. As the kernel of the unit operatoris the Dirac delta-function δ4(x′ − x), the Green function is a fundamentalsolution of the non-homogeneous equation

Lu(x) = f(x), (4.1)

8Both coincide up to notation or a meaning of the corresponding symbols.


i.e. treated as a generalized function g(x′, x) is a solution of

Lx′g(x′, x) = δ4(x′ − x). (4.2)

Given the Geen function g(x′, x), the solution of (4.1) is

u(x′) =

∫g(x′, x)f(x) d4x. (4.3)

A concrete Green function g(x′, x) for L (or (4.1)) satisfies, besides (4.2),certain (homogeneous) boundary conditions on x′ with fixed x, i.e. it is thesolution of a fixed boundary-value problem for equation (4.2). Hence, ifgf (x′, x) is some fundamental solution, then

g(x′, x) = gf (x′, x) + g0(x′, x) (4.4)

where g0(x′, x) is a solution of the homogeneous equation Lx′g0(x

′, x) = 0chosen such that g(x′, x) satisfies the required boundary conditions.

Suppose g(x′, x) is a Green function of L for some boundary-value (orinitial-value) problem. Then, using (4.2), we can verify that

Lx′

(∫g(x′, x)u(x) d3x

)= δ(ct′ − ct)u(x′) (4.5)

where x = (ct,x) and x′ = (ct′,x′). Therefore the solution of the problem

Lxu(x) = 0, u(ct0,x) = u0(x) (4.6)

is

u(x) =

∫g(x, (ct0,x0)

)u0(x0) d3x0, x0 = (ct0,x0) for t 6= t0. (4.7)

From here we can make the conclusion that, if (4.6) admits an evolutionoperator U such that (cf. (I.2.2))

u(x) ≡ u(ct,x) = U(t, t0)u(ct0,x) (4.8)

or (cf. (2.2))

u(x) = U(x, x0)u(x0), (4.9)

then the r.h.s of (4.7) realizes U as an integral operator with a kernel equalto the Green function g.

Since all (relativistic or not) wave equations are versions of (4.6), thecorresponding evolution operators, if any, and Green functions (propaga-tors) are connected as just described. Moreover, if some wave equation doesnot admit (directly) evolution operator, e.g. if it is of order greater than


one, then we can define it as the corresponding version of the integral op-erator in the r.h.s. of (4.7). In this way is established a one-to-one ontocorrespondence between the evolution operators and Green functions forany particular problem like (4.6).

And a last general remark. The so-called S-matrix finds a lot of applica-tions in quantum theory [4,6,14]. By definition S is an operator transformingthe system’s state vector ψ(−∞,x) before scattering (reaction) into the oneψ(+∞,x) after it:

limt→+∞

ψ(ct,x) =: S limt→−∞

ψ(ct,x).

So, e.g. when (4.8) takes place, we have

S = limt±→±∞

U(t+, t−) =: U(+∞,−∞). (4.10)

Thus the above-mentioned connection between evolution operators andGreen functions can be used for expressing the S-matrix in terms of prop-agators. Such kind of formulae are often used in relativistic quantum me-chanics [6].

4.2. Nonrelativistic case (Schrodinger equation)

The (retarded) Green function g(x′, x), x′, x ∈M , for the Schrodinger equa-tion (I.2.1) is defined as the solution of the boundary-value problem [6, § 22]

[i~∂

∂t′−H(x′)

]g(x′, x) = δ4(x′ − x), (4.11)

g(x′, x) = 0 for t′ < t (4.12)

where x′ = (ct′,x′) x = (ct,x), H(x) is system’s Hamiltonian, and δ4(x′−x)is the 4-dimensional (Dirac) δ-function.

Given g, the solution ψ(x′) (for t′ > t) of (I.2.1) is9

θ(t′ − t)ψ(x′) = i~

∫d3xg(x′, x)ψ(x) (4.13)

where the θ-function θ(s), s ∈ R is defined by θ(s) = 1 for s > 0 andθ(s) = 0 for s < 0.

9One should not confuse the notation ψ(x) = ψ(ct,x), x = (ct,x) of this section andψ(t) from Sect. I.2. The latter is the wavefunction at a moment t and the former is itsvalue at the spacetime point x = (ct,x). Analogously, Ψγ(x) ≡ Ψγ(ct,x) := l−1

γ(t)

(ψ(ct,x)

)

should not be confused with Ψγ(t) from Sect. I.2. A notation like ψ(t) and Ψγ(t) will beused if the spatial parts of the arguments is inessential, as in Sect. I.2, and there is no riskof ambiguities.


Combining (I.2.2) and (4.13), we find the basic connection between theevolution operator U and Green function of Schrodinger equation:

θ(t′ − t)[U(t′, t)

(ψ(ct,x′)

)]= i~

∫d3xg

((ct′,x′), (ct,x)

)ψ(ct,x). (4.14)

Actually this formula, if g is known, determines U(t′, t) for all t′ and t,not only for t′ > t, as U(t′, t) = U−1(t, t′) and U(t, t) = idF with F beingthe system’s Hilbert space (see (I.2.2) or [7, sect. 2]). Consequently theevolution operator for the Schrodinger equation can be represented as anintegral operator whose kernel up to the constant i~ is exactly the (retarded)Green function for it.

To write the bundle version of (4.14), we introduce the Green operatorwhich is simply a multiplication with the Green function:

G(x′, x) := g(x′, x) idF : F → F . (4.15)

The corresponding to it Green morphism G is given via (see (I.2.11) andcf. (I.2.4))

Gγ(x′, x) := l−1γ(t′) ◦ G(x′, x) ◦ lγ(t) = g(x′, x)l−1

γ(t′) ◦ lγ(t) : Fγ(t) → Fγ(t′).

(4.16)

Now, acting on (4.14) from the left by l−1γ(t′)

and using (I.2.3) and (I.2.4),we obtain

θ(t′ − t)[Uγ(t′, t)

(Ψγ(ct,x′)

)]= i~

∫d3xGγ

((ct′,x′), (ct,x)

)Ψγ(ct,x).

(4.17)

Therefore for the Schrodinger equation the the evolution transport U can berepresented as an integral operator with kernel equal to i~ times the Greenmorphism G.

Taking as a starting point (4.14) and (4.17), we can obtain different rep-resentations for the evolution operator and transport by applying concreteformulae for the Green function. For example, if a complete set {ψa(x)} oforthonormal solutions of Schrodinger equation satisfying the completenesscondition10

∑

a

ψa(ct,x′)ψ∗

a(ct,x) = δ3(x′ − x)

is know, then [6, § 22]

g(x′, x) =1

i~θ(t′ − t)

∑

a

ψa(x′)ψ∗

a(x)

10Here and below the symbol∑

a denotes a sum and/or integral over the discrete and/orcontinuous spectrum. The asterisk (∗) means complex conjugation.


which, when substituted into (4.14), implies

U(t′, t)ψ(ct,x′) =∑

a

ψa(x′)

∫d3 xψ∗

a(ct,x)ψ(ct,x). (4.18)

Note, the integral in this equation is equal to the a-th coefficient of theexpansion of ψ over {ψa}.

4.3. Dirac equation

Since the Dirac equation (2.1) is a first-order linear partial differential equa-tion, it admits both evolution operator and Green function(s) (propaga-tor(s)). From a generic view-point, the only difference from the Schrodingerequation is that (2.1) is a matrix equation; so the corresponding Greenfunctions are actually Green matrices, i.e. Green matrix-valued functions.Otherwise the results of Subsect. 4.2 are mutatis mutandis transferred intothe theory of Dirac equation.

The (retarded) Green matrix (function) or propagator for Dirac equa-tion (2.1) is a 4× 4 matrix-valued function g(x′, x) depending on two argu-ments x′, x ∈M and such that [4, sect. 2.5.1 and 2.5.2]

(i~D/x′ −mc114)g(x′, x) = δ(x′ − x) (4.19)

g(x′, x) = 0 for t < t′. (4.20)

For a free Dirac particle, i.e. for D/ = ∂/ or Aµ = 0, the explicit expressiong0(x

′, x) for g(x′, x) is derived in [4, sect. 2.5.1], where the notation K insteadof g0 is used. In an external electromagnetic field Aµ the Green matrix g isa solution of the integral equation11

g(x′, x) = g0(x′, x) +

∫d4y g0(x

′, y)e

cA/(y)g(y, x) (4.21)

which includes the corresponding boundary condition.12 The iteration ofthis equation results in the perturbation series for g (cf. [4, sect. 2.5.2]).

If the (retarded) Green matrix g(x′, x) is known, the solution ψ(x′) ofDirac equation (for t′ > t) is

θ(t′ − t)ψ(x′) = i~

∫d3x g(x′, x)γ0ψ(x). (4.22)

11The derivation of (4.21) is the same as for the Feynman propagators SF and SA givenin [4, sect. 2.5.2]. The propagators SF and SA correspond to g0 and g respectively, butsatisfy other boundary conditions [4,6].

12Due to (4.4), the integral equation (4.21) is valid for any Green function (matrix) ofthe Dirac equation (2.1).


Hence, denoting by U and U the non-relativistic (see Sect. I.4) and rela-tivistic (see Sect. 2), respectively, Dirac evolution operators, from the equa-tions (I.2.2), (2.2), and (4.22), we find:

θ(t′ − t)[U(t′, t)

(ψ(ct,x′)

)]= θ(t′ − t)

[U(x′, x)

(ψ(x)

)]

= i~

∫d3x g

((ct′,x′), (ct,x)

)γ0ψ(ct,x). (4.23)

So, in both cases the evolution operator admits an integral representationwhose kernel up to the right multiplication with i~γ0 is equal to the (re-tarded) Green function for Dirac equation.

Similarly to (4.17), now the bundle version of (4.23) is

θ(t′ − t)[U (t′, t)

(Ψγ(ct,x′)

)]= i~

∫d3xGγ(x′, x)G0(γ(t))Ψγ(ct,x),

(4.24a)

θ(t′ − t)[U (x′, x)

(Ψ(x)

)]= i~

∫d3xG(x′, x)G0(x)Ψ(x) (4.24b)

where G0(x) is defined in (2.18) for µ = 0 and (cf. (4.16))

Gγ(x′, x) := l−1γ(t′) ◦ G(x′, x) ◦ lγ(t), G(x′, x) := l−1

x′ ◦ G(x′, x) ◦ lx (4.25)

with (cf. . (4.15))

G(x′, x) = g(x′, x) idF . (4.26)

(Here F is the space of 4-spinors.)Analogously to the above results, one can obtained such for other prop-

agators, e.g. for the Feynman one [4,6], but we are not going to do this hereas it is a trivial variant of the procedure described.

4.4. Klein-Gordon equation

The (retarded) Green function g(x′, x) for the Klein-Gordon equation (3.1)is a solution to the boundary-value problem [4, sect. 1.3.1]

(DµDµ +

m2c2

~2

)∣∣∣x′g(x′, x) = δ4(x′ − x), (4.27)

g(x′, x) = 0 for t′ < t. (4.28)

For a free particle its explicit form can be found in [4, sect. 1.3.1].A simple verification proves that, if g(x′, x) is known, the solution φ

of (3.1) (for t′ > t) is given by (x0 = ct)

θ(t′ − t)φ(x′) =

∫d3x

[∂g(x′, x)∂x0

φ(x) + g(x′, x)(2∂φ(x)

∂x0−

e

i~cA0(x)φ(x)

)].

(4.29)


Introducing the matrices

ψ(x) :=

(φ(x)

∂0|xφ(x)

), g(x′, x) :=

(D0|xg(x

′, x)2g(x′, x)

), (4.30)

we can rewrite (4.29) as

θ(t′ − t)φ(x′) =

∫d3x g⊤(x′, x) · ψ(x) (4.31)

where the dot (·) denotes matrix multiplication.An important observation is that for ψ the Klein-Gordon equation trans-

forms into first-order Schrodinger-type equation (see Sect. I.5) with Ha-miltonian K−G

cH given by (I.5.3) in which id... is replaced by c id....Denoting the (retarded) Green function, which is in fact 2 × 2 matrix,

and the evolution operator for this equation by G(x′, x) =[Ga

b(x′, x)

]2

a,b=1

and U(t′, t) =[Ua

b(t′, t)

]2

a,b=1, respectively, we see that (cf. Subsect. 4.2,

equation (4.13))

θ(t′ − t)[U(t′, t)ψ(ct,x)

]= θ(t′ − t)ψ(x′) =

∫d3x G(x′, x)ψ(x). (4.32)

Comparing this equations with (4.30) and (4.31), we find

(G1

1(x′, x), G1

2(x′, x)

)= g⊤(x′, x).

The other matrix elements of G can also be connected with g(x′, x) and itsderivatives, but this is inessential for the following.

In this way we have connected, via (4.32), the evolution operator and the(retarded) Green function for a concrete first-order realization of Klein-Gor-don equation. It is almost evident that this procedure mutatis mutandisworks for any such realization; in every case the corresponding Green func-tion (resp. matrix) being a (resp. matrix-valued) function of the Green func-tion g(x′, x) introduce via (4.27) and (4.28). For instance, the treatment ofthe 5-dimensional realization given by (3.2) and (3.3) is practically identicalto the one of Dirac equation in Subsect. 4.3, only the γ-matrices γµ have tobe replace with the 5× 5 matrices Γµ (defined by (3.2)). This results into a5 × 5 matrix evolution operator U(x′, x), etc.

Since the bundle version of (4.32) or an analogous result for U(x′, x) isabsolutely trivial (cf. Subsect. 4.2 and 4.3 resp.), we are not going to writeit here; up to the meaning of notation it coincides with (4.17) or (4.24b)respectively.

At the end, we notice that via (4.29) can be defined an ‘evolution oper-ator’ φ(x) 7→ φ(x′) (or φ(ct,x) 7→ φ(ct′,x)), but such a map does not have‘good’ properties and does not find applications.


5. Conclusion

In this investigation we have reformulated the relativistic wave equations interms of fibre bundles. In the bundle formulation the wavefunctions are rep-resented as (state) liftings of paths or sections along paths (time-dependentapproach) or simply sections (covariant approach) of a suitable vector bundleover the spacetime. The covariant approach, developed in the present work,has an advantage of being explicitly covariant while in the time-dependentone the time plays a privilege role. In both cases the evolution (in time orin spacetime resp.) is described via an evolution transport which is a lineartransport (along paths or along the identity map of the spacetime resp.)in the bundle mentioned. The state liftings or sections are linearly trans-ported by means of the corresponding evolution transports. An equivalentversion of this statement is the new bundle form of the wave equations: thederivation(s) assigned to the evolution transports of the corresponding statesections identically vanish. We also have explored some links between evolu-tion operators or transports and the retarded Green functions (or matrices)for the corresponding wave equations: the former turn to have realizationas integral operators whose kernel is equal to the latter ones up to a multi-plication with a constant complex number or matrix.

These connections suggest the idea for introducing ‘retarded’, or, in asense, ‘causal’ evolution operators or transports as a product of the evolu-tion operators or transports with θ-function of the difference of the timescorresponding to the first and second arguments of the transport or operator.

A further development of the ideas presented in this investigation natu-rally leads to their application to (quantum) field theory which will be doneelsewhere.

Appendix A. Linear transports along maps in

fibre bundles

In this appendix we recall a few simple facts concerning (linear) transportsalong maps, in particular along paths, required for the present investigation.The below-presented material is abstracted from [11–13] where further de-tails can be found (see also [7, sect. 3]).

Let (E, π,B) be a topological bundle with base B, bundle (total, fibre)space E, projection π : E → B, and homeomorphic fibres π−1(x), x ∈ B.Let the set N be not empty, N 6= ∅, and there be given a map κ : N → B.By idX is denoted the identity map of a set X.

Definition A.1. A transport along maps in the bundle (E, π,B) is a mapK assigning to any map κ : N → B a map Kκ, transport along κ, such


that Kκ : (l,m) 7→ Kκ

l→m, where for every l,m ∈ N the map

Kκ

l→m : π−1(κ(l)) → π−1(κ(m)), (A.1)

called transport along κ from l to m, satisfies the equalities:

Kκ

m→n ◦Kκ

l→m = Kκ

l→n, l,m, n ∈ N, (A.2)

Kκ

l→l = idπ−1(κ(l)), l ∈ N. (A.3)

If (E, π,B) is a complex (or real) vector bundle and the maps (A.1) arelinear, i.e.

Kκ

l→m(λu+ µv) = λKκ

l→mu+ µKκ

l→mv, λ, µ ∈ C (or R), u, v ∈ π−1(κ(l)),(A.4)

the transport K is called linear. If κ belongs to the set of paths in B,κ ∈ {γ : J → B, J being R-interval}, we said that the transport K is alongpaths.

For the present work is important that the class of linear transports alongthe identity map idB of B coincides with the class of flat linear transportsalong paths13 (see the comments after equation (2.3) of [11]).

The general form of a transport along maps is described by the followingresult.

Theorem A.1. Let κ : N → B. The map K : κ 7→ Kκ : (l,m) 7→ Kκ

l→m,l,m ∈ N is a transport along κ if and only if there exist a set Q and afamily of bijective maps {Fκ

n : π−1(κ(n)) → Q, n ∈ N} such that

Kκ

l→m = (Fκ

m)−1 ◦ (Fκ

l ) , l,m ∈ N. (A.5)

The maps Fκn are defined up to a left composition with bijective map depend-

ing only on κ, i.e. (A.5) holds for given families of maps {Fκn : π−1(κ(n)) →

Q, n ∈ N} and {′Fκn : π−1(κ(n)) → ′Q, n ∈ N} for some sets Q and ′Q iff

there is a bijective map Dκ : Q→ ′Q such that

′Fκ

n = Dκ ◦ Fκ

n , n ∈ N. (A.6)

For the purposes of this investigation we need a slight generalizationof [11, definition 4.1], viz we want to replace in it N ⊂ R

k with an arbitrarydifferentiable manifold. Let N be a differentiable manifold and {xa : a =1, . . . ,dimN} be coordinate system in a neighborhood of l ∈ N . For ε ∈(−δ, δ) ⊂ R, δ ∈ R+ and l ∈ N with coordinates la = xa(l), we define

13The flat linear transports along paths are defined as ones with vanishing curvatureoperator [17, sect. 2]. By [17, theorem. 6.1] we can equivalently define them by theproperty that they depend only on their initial and final points, i.e. if Kγ

s→t, γ : J → B,s, t ∈ J ⊂ R, depends only on γ(s) and γ(t) but not on the path γ itself.


lb(ε) ∈ N , b = 1, . . . ,dimN by lab (ε) := xa(lb(ε)) := la + εδab where the

Kroneker δ-symbol is given by δab = 0 for a 6= b and δa

b = 1 for a = b. Letξ = (E, π,B) be a vector bundle, κ : N → B be injective (i.e. 1:1 mapping),and Secp(ξ) (resp. Sec(ξ)) be the set of Cp (resp. all) sections over ξ. LetLκ

l→m be a C1 (on l) linear transport along κ. Now the modified definitionreads:14

Definition A.2. The a-th, 1 ≤ a ≤ dimN partial (section-)derivationalong maps generated by L is a map aD : κ 7→ aD

κ where the a-th (partial)derivation aD

κ along κ (generated by L) is a map

aDκ : l 7→ Dκ

la , (A.7)

where the a-th (partial) derivative Dκ

la along κ at l is a map

Dκ

la : Sec1(ξ|

κ(N)

)→ π−1(κ(l)) (A.8)

defined for σ ∈ Sec1(ξ|κ(N)

)by

(aDκσ) (κ(l)) := lim

ε→0

[1

ε

(Lκ

la(ε)→lσ(κ(la(ε))) − σ(κ(l)))]. (A.9)

Accordingly can be modified the other definitions of [11, sect. 4], all theresults of it being mutatis mutandis valid. In particular, we have:

Proposition A.1. The operators Dκ

la are (C-)linear and

Dκ

ma ◦ Lκ

l→m ≡ 0. (A.10)

Proposition A.2. If σ ∈ Sec1(ξ|κ(N)

), then

Dκ

laσ =∑

i

[∂σi(κ(l))

∂la+

∑

j

aΓij(l; κ)σj(κ(l))

]ei(l), (A.11)

where {ei(l)} is a basis in π−1(κ(l)), σ(κ(l)) =:∑

i σi(κ(l))ei(l), and the

coefficients of L are defined by

aΓij(l; κ) :=

∂Lij(l,m; κ)

∂ma

∣∣∣∣∣m=l

= −∂Li

j(m, l; κ)

∂ma

∣∣∣∣∣m=l

. (A.12)

Here Lji(· · · ) are the components of L, Lκ

l→mei(l) =:∑

j Lji(m, l; κ)ej(m).

The above general definitions and results will be used in this work in thespecial case of linear transports along the identity map of the bundle’s base.

14We present below, in definition A.2, directly the definition of a section-derivation alonginjective mapping κ as only it will be employed in the present paper (for κ = idN). If κ isnot injective, the mapping (A.7) could be multiple-valued at the points of self-intersectionof κ, if any. For some details when κ is an arbitrary path in N , see [7, subsect. 3.3].


Appendix B. Matrix operators

In this appendix we point to some peculiarities of linear (matrix) operatorsacting on n×1, n ∈ N, matrix fields over the space-time M . Such operatorsnaturally appear in the theory of Dirac equation where one often meets 4×4matrices whose elements are operators; e.g. an operator of this kind is ∂/ :=

γµ∂µ =[(γµ)αβ∂µ

]3

α,β=0where γµ are the well known Dirac γ-matrices [4,6].

We call an n×n, n ∈ N matrix B = [bαβ]nα,β=1 a (linear) matrix operator15

if bαβ are (linear) operators acting on the space K 1 of C1 functions f : M →

C. If {f0ν } is a basis in the set M(n, 1) of n × 1 matrices with the µ-th

element of f0ν being (f0

ν )µ := δµν ,16 then by definition

Bψ := B(ψ) := B · (ψ) :=n∑

α,β=1

(bαβ(ψβ

0

)f0

α, ψ = ψβ0 f

0β ∈M(n, 1).

(B.1)

For instance, we have ∂/ψ =∑3

α,β,µ=0(γµ)αβ(∂µψ

β0 )f0

α.To any constant matrix C = [cαβ], cαβ ∈ C, corresponds a matrix opera-

tor C := [cαβ idK 1]. Since Cψ ≡ Cψ for any ψ, we identify C and C and willmake no difference between them.

The multiplication of matrix operators, denoted by ⊙, is a combinationof matrix multiplication (denoted by ·) and maps (operators) composition(denoted by ◦). If A = [aα

β] and B = [bαβ] are matrix operators, the productof A and B is also a matrix operator such that

AB := A⊙B :=

[∑

µ

aαµ ◦ bµβ

]. (B.2)

One can easily show that this is an associative operation. It is linear in thefirst argument and, if its first argument is linear matrix operator, then itis linear in its second argument too. For constant matrices (see above) themultiplication (B.2) coincides with the usual matrix one.

Let {fα(x)} be a basis in M(n, 1) depending on x ∈ M and f(x) :=

[fβα (x)] be defined by the expansion fα(x) = fβ

α (x)f0β . The matrix of a

matrix operator B = [bαβ ] with respect to {fα}, i.e. the matrix of the matrixelements of B considered as an operator, is also a matrix operatorB := [Bα

β]such that

Bψ|x =:∑

α,β

(Bα

β(ψβ))∣∣

xfα(x), ψ(x) = ψβ(x)fβ(x). (B.3)

15Also a good term for such an object is (linear) matrixor.16δµ

ν = 0 for µ 6= ν and δµν = 1 for µ = ν. From here to equation (B.10) in this appendix

the Greek indices run from 1 to n ≥ 1.


Therefore

ϕ = Bψ ⇐⇒ ϕ = Bψ, (B.4)

where ψ ∈M(n, 1) is the matrix of the components of ψ in the basis given.Comparing (B.1) and (B.3), we get

Bαβ =

∑

µ,ν

(f−1(x)

)α

µbµν ◦

(f ν

β (x) idK 1

). (B.5)

For any matrix C = [cαβ] ∈ GL(n,C), considered as a matrix operator(see above), we have

C =[Cα

β(x)], Cα

β(x) =(f−1(x)

)α

µcµνf

νβ , C(fβ) = Cα

βfα. (B.6)

Therefore, as one can expect, the matrix of the unit matrix 11n =[δβα

]n

α,β=1is exactly the unit matrix:

111111n = 11n. (B.7)

If {fµ(x)} does not depend on x, i.e. if fβα are complex constants, and bµν

are linear, than (B.5) implies

Bαβ =

(f−1(x)

)α

µf ν

β (x)bµν for ∂fβα (x)/∂xµ = 0. (B.8)

In particular, for a linear matrix operator B, we have

B = B for fαβ = δα

β (i.e. for fα(x) = f0α). (B.9)

Combining (B.2) and (B.5), we deduce that the matrix of a product ofmatrix operators is the product of the corresponding matrices:

C = A⊙B ⇐⇒ C = AB = A⊙B. (B.10)

After a simple calculation, we find the matrices of 114∂µ and ∂/ = γµ∂µ:17

∂µ = 114∂µ + Eµ(x), ∂/ = Gµ(x)(114∂µ + Eµ(x)

)= Gµ(x)∂µ (B.11)

where Gµ(x) =[Gλµ

α (x)]3

α,λ=0and Eµ(x) =

[Eλ

µα(x)]3

α,λ=0are defined via the

expansions

γµfν(x) =: Gλµν (x)fλ(x), ∂µfν(x) =: Eλ

µν(x)fλ(x). (B.12)

So, Gµ is the matrix of γµ considered as a matrix operator (see (B.6)).18

Evidently

∂/ = ∂/ for fα(x) = f0α. (B.13)

17Here and below the Greek indices run from 0 to 3.18We denote the matrix of γµ by Gµ instead of γµ(x) as usually [4, 6] by γ is denoted

the matrix 3-vector (γ1, γ2, γ3).


Combining (B.6),(B.7), and (B.11), we get that the matrix of the matrixoperator i~D/−mc114, entering in the Dirac equation (2.1), is

i~D/−mc1111114 = i~Gµ(x)(114Dµ + Eµ(x)

)−mc114, (B.14)

so that

i~D/ −mc1111114 = i~D/−mc114 for fα(x) = f0α. (B.15)

References

[1] Bozhidar Z. Iliev. Fibre bundle formulation of relativistic quantummechanics. I. Time-dependent approach. LANL arXiv server, E-printNo. quant-ph/0105056, May 2001.

[2] Bozhidar Z. Iliev. Fibre bundle formulation of nonrelativistic quantummechanics. V. interpretation, summary, and discussion. (LANL arXivserver, E-print No. quant-ph/9902068), 1999.

[3] Bozhidar Z. Iliev. Fibre bundle formulation of nonrelativistic quan-tum mechanics (full version). LANL arXiv server, E-print No. quant-ph/0004041, April 2000.

[4] C. Itzykson and J.-B. Zuber. Quantum field theory. McGraw-Hill BookCompany, New York, 1980.

[5] A. M. L. Messiah. Quantum mechanics, volume II. North Holland,Amsterdam, 1962.

[6] J. D. Bjorken and S. D. Drell. Relativistic quantum mechanics, vol-ume 1. McGraw-Hill Book Company, New York, 1964.

[7] Bozhidar Z. Iliev. Fibre bundle formulation of nonrelativistic quantummechanics. I. Introduction. The evolution transport. Journal of PhysicsA: Mathematical and General, 34(23):4887–4918, 2001. LANL arXivserver, E-print No. quant-ph/9803084.

[8] A. M. L. Messiah. Quantum mechanics, volume I. North Holland,Amsterdam, 1961.

[9] Bozhidar Z. Iliev. Linear transports along paths in vector bundles.II. Some applications. JINR Communication E5-93-260, Dubna, 1993.

[10] Bozhidar Z. Iliev. Fibre bundle formulation of nonrelativistic quantummechanics. II. Equations of motion and observables. Journal of PhysicsA: Mathematical and General, 34(23):4919–4934, 2001. LANL arXivserver, E-print No. quant-ph/9804062.


[11] Bozhidar Z. Iliev. Transports along maps in fibre bundles. JINR Com-munication E5-97-2, Dubna, 1997. (LANL arXiv server, E-print No.dg-ga/9709016).

[12] Bozhidar Z. Iliev. Normal frames and linear transports along paths invector bundles. LANL arXiv server, E-print No. gr-qc/9809084, 1998.

[13] Bozhidar Z. Iliev. Linear transports along paths in vector bundles.I. General theory. JINR Communication E5-93-239, Dubna, 1993.

[14] N. N. Bogolyubov and D. V. Shirkov. Introduction of the theory ofquantized fields. Wiley, New York, 1980. (Translation from the thirdRussian ed.: Nauka, Moscow, 1976).

[15] N. F. Nelipa. Physics of elementary particles. Vyshaya shkola, Moscow,1977. (In Russian).

[16] Physical encyclopedia, chief editor Prohorov A. M., volume 1, Moscow,1988. Sovetskaya Entsiklopediya (Soviet encyclopedia). (In Russian).

[17] Bozhidar Z. Iliev. Linear transports along paths in vector bundles. V.Properties of curvature and torsion. JINR Communication E5-97-1,Dubna, 1997. (LANL arXiv server, E-print No. dg-ga/9709017).

ii. covariant approach - pennsylvania state university

Documents