aa r X i v : . [ phy s i c s . c l a ss - ph ] A ug The Bones of Sophus Lie
Clinton L. Lewis
Division of Science and Mathematics,West Valley College, Saratoga, California (retired) ∗ (Dated: August 5, 2020)The gauge covariant derivative of a wave function is ubiquitous in gaugetheory, and with associated gauge transformations it defines charged currentsinteracting with external fields, such as the Lorentz force exerted by an elec-tromagnetic field. It is the gauge covariant derivative which defines how anexternal field acts upon the wave function. This paper constructs the gaugecovariant derivative, then uses the elegant framework of Lagrangian mechanicsto derive two “divergence equations” from a general Lagrangian, one applyingto the charged current, the other to energy-momentum. The student will ap-preciate the construction of the gauge covariant derivative of a classical wavefunction using only matrices, linear transformations, external fields, and par-tial derivatives. More unusual is using the principle of covariance, rather thangroup theory as guidance in the construction, but with exactly the same result.Advantage is taken of the close analogy with coordinate covariance of tensors.The details of deriving these two divergences provides motivation and a pathto understanding the gauge covariant derivative, the underlying non-abelianLie algebra, its application to building Lagrangians, and the resulting defini-tions of charged current and energy-momentum. All results are generalized tocurved space-time. CONTENTS
I. Introduction 4II. Continuity equations and conservation 6III. Two divergences, the abelian case 7IV. The Lagrangian for a wave function 10V. The gauge transformation of the wave function 10VI. Transformation properties of the gauge covariant derivative 13VII. Construction of the gauge covariant derivative 14VIII. Parameterization of the transformation 16IX. Taylor expansion of the transformation 17X. Representation of the gauge transformation matrix 19XI. Infinitesimal gauge transformation 19XII. Gauge invariant generators, Sophus Lie’s second theorem 21XIII. Vanishing derivative of the generators 23XIV. Vectors in parameter space 25XV. Definition of the field strength tensor 25XVI. Definition of the Cartan-Killing metric 26XVII. Invariant structure constants, or the Jacobi identity 28XVIII. The gauge transformation of the Lagrangian 28XIX. The divergence of the charged current 29XX. The Divergence of the energy-momentum tensor 31XXI. The generalized Lorentz Force 33XXII. Vanishing divergence may not imply local continuity 33XXIII. Conclusion 34Acknowledgments 35A. Example: the wave equation in a gauge field 35B. Example: the current 39C. Example: Yang-Mills equations of motion 40D. Definition of the Tensor Dual 42References 43
I. INTRODUCTION
We study how a classical wave function, its gauge covariant derivative,and the corresponding Lagrangian respond to a gauge transformation, thenconspire to define a charged current, the divergence of that current, and an ex-ternal Lorentz force acting upon the energy-momentum of the system. Theseresults appear as two “divergence-type” equations, one for charge, the otherfor energy-momentum.These are ancient topics in gauge theory , and Lagrangian mechanics, butthe presentation here is shorted by using the principle of covariance as thecentral guiding principle. Covariance is the response of the wave function to alinear homogeneous transformation. The object of this paper is to constructfrom first principles an essential tool in building Lagrangians, the gauge co-variant derivative of a classical wave function in curved space-time. In thesimplest case only two objects enter the Lagrangian, the wave function andits derivative.The hope is to appeal to students with a presentation of the mathematicalproperties of the gauge covariant derivative, but closely motivated by physicsin the form of the two divergences, and using only elements of differentialequations, matrix linear algebra, and tensor index notation.The gauge covariant derivative appears in Lagrangians which are invariantunder the application of a gauge transformation. Such a Lagrangian is said tohave a symmetry with respect to this transformation. Further, this symmetryimplies a charged current with a vanishing divergence. In the end, since allparts defined here have specific gauge transformation properties, the result isa toolbox for building gauge invariant Lagrangians.Consider a wave function represented as a column matrix φ with complex-valued elements, each a function of position in space-time x µ , µ = 0 , , , φ ( x ) = φ (1) ( x )... φ ( n ) ( x ) (1)The partial derivative is defined according to the usual limiting process, ∂ φ ( x ) ∂x µ = lim ∆ x µ → φ ( x + ∆ x µ ) − φ ( x )∆ x µ (2)The partial derivative applies to each component of the wave function. ∂ φ ( x ) ∂x µ = ∂φ ( x ) ∂x µ ... ∂φ n ( x ) ∂x µ (3)A gauge covariant derivative of a wave function is related to the partialderivative, but adds an additional conceptual layer – that of the linear andhomogeneous response of the wave function to a gauge transformation . Byconstruction, the gauge covariant derivative of the wave function responds tothe same gauge transformation.Implement the gauge transformation by left-multiplying the wave functionby the transformation represented as a square matrix T ( x ), T φ (4)By construction to be explored, the gauge covariant derivative of the wavefunction D µ ( φ ) covaries which means that the derivative of the wave functionresponds to the identical transformation as the wave function itself. T D µ ( φ ) (5)The gauge transformation matrix, to be defined later, is essential to the defi-nition of the gauge covariant derivative.How to motivate and add linear gauge transformation properties to theordinary partial derivative to construct a gauge covariant derivative, occupiesthe majority of this paper.Analogously , the coordinate covariant derivative used in tensor algebrahas linear homogeneous transformation properties under a general coordinatetransformation. Historically, general relativity with essential use of coordinatecovariant derivatives was introduced in 1915, while gauge transformations wereintroduced later by Hermann Weyl, and Kibble , and still later as part of theStandard Model.Two applications of the gauge covariant derivative are explored here, • the definition and divergence of a charged current vector , and • the definition and divergence of the energy-momentum tensor while sub-ject to external Lorentz forces.These “two divergences” provide physical motivation for the mathematicalproperties and construction of the gauge covariant derivative.The stage for the evolution of fields in our system is a general space-timemetric. The metric, although arbitrary, is a fixed background metric so thatEinstein equations do not apply. Two fields exist in this space-time, a wavefunction representing matter, and a generalization of the electromagnetic fieldacting as an external force.The notation emphasizes covariance, so tensor notation, Einstein sum-mation and covariant derivatives are used throughout. The development,although covariant, is not general relativistic since the system considered hereis subject to a force due to an external gauge field, a situation impossible ingeneral relativity where there are no “external” forces. Matrix notation is used to avoid a paroxysm of indices.
Bold type indicates a matrix with accompanying gauge transformation properties.Derivatives with respect to a matrix are to be evaluated component bycomponent, with matrix multiplication indicating summation of matrix com-ponents. A matrix equation may be converted to a component equation byrestoring matrix indices. A partial derivative of an invariant scalar, such asthe Lagrangian, with respect to a column matrix yields a row matrix.
II. CONTINUITY EQUATIONS AND CONSERVATION
Continuity equations have intuitive appeal since continuity conveys thefeeling of an enduring, ponderable substance, neither created nor destroyedin an isolated system. These continuity equations appear as a vanishing di-vergence of a vector, hence, our pursuit of the two divergences mentionedabove.The continuity equation for a fluid of density P and current J at a pointin space ( r, t ) is, ∂∂t P ( r, t ) + ∇ · J ( r, t ) = 0 (6)The continuity equation implies a conservation law in the sense that the quan-tity Q of fluid contained in a given volume V with an exterior surface S variesonly with the passage of current J through the exterior surface. ∂∂t Q = ∂∂t Z V P ( r, t ) dV = − Z S J ( r, t ) · d S (7)If the surface is extended, perhaps to infinity, to include all external currents,then the quantity in the volume is constant, so that the derivative with respectto time would be zero. Fluid conservation means that no fluid is created orlost within the volume, without having entered or exited through the enclosingsurface. The vanishing “divergence-type” expression in Eq. (6) becomes thefollowing in Minkowski space-time with a diagonal constant metric. ∂ µ j µ = 0 (8)The density P is the time component of the 4 dimensional current vector j µ .In covariant tensor notation, the vanishing divergence represents a continuityequation for a current in a general curved space-time. ∇ µ j µ = 0 (9)The vanishing covariant derivative is brought into a form exposing ordinarypartial derivatives by using an identity . The absolute value of the determi-nant of the metric g µν ( x ) is “ g ” in the following tensor expression. This formmakes clear the continuity equation consisting of the vanishing divergence ofthe ordinary partial derivative. ∇ µ j µ = 1 √ g ∂ µ ( √ gj µ ) = 0 (10)This form of the vanishing covariant divergence is a generalization of Eq. (3),and is again an equation of continuity because of the appearance of the ordi-nary partial derivative. Hence the importance of discussing vanishing diver-gence of current and energy momentum, and the connection to conservation.The pursuit of this type of “local” conservation law at a point, or continuityequation is one of the motivations for our study of gauge covariant derivativesappearing in a vanishing divergence. III. TWO DIVERGENCES, THE ABELIAN CASE
As an example of the two divergences mentioned above, we explore thewave function in an electromagnetic field. These divergence equations arelater generalized by using a more general gauge covariant derivative. In thefollowing, ~ = 1.The prototypical gauge transformation is that of electromagnetics and thecorresponding phase transformation of the wave function. This exampleprovides guidance for the more complex non-abelian gauge transformationsexplored here.The gauge covariant derivative of the single-component, complex-valuedwave function D µ φ ( x ), a function of space-time position x µ , is the familiar, D µ φ = ∂ µ φ + ieA µ φ (11)where A µ is the electromagnetic 4-vector potential, and where “ e ” is the ordi-nary electric charge coupling constant. This defines a gauge covariant deriva-tive, because the derivative has the same phase (gauge) transformation prop-erties as the wave function field itself. The derivative “covaries” with respectto a phase change indicated by the real parameter θ ( x ) as follows, φ −→ G e ieθ ( x ) φD µ φ −→ G e ieθ ( x ) D µ φ (12)where the electromagnetic vector potential must simultaneously gauge trans-form in correspondence with the phase transformation of the wave function. A µ −→ G A µ − ∂ µ θ (13)The right arrow “ −→ G ” indicates “replacement” since these are finite transfor-mations. Presented in derivative form, the infinitesimal version of the samegauge transformation is δφ/δθ = ieφδD µ φ/δθ = ieD µ φδA µ = − ∂ µ ( δθ ) (14)where δ is the transformation due to a small change in the parameter δθ .The gauge covariant derivative D µ does not commute in general. Definethe antisymmetric electromagnetic field tensor F µν . F µν = ∂ µ A ν − ∂ ν A µ (15)which is gauge transformation invariant , F µν −→ G F µν (16)Computation shows that the commutator of the gauge covariant derivativeis [ D µ , D ν ] φ = ieF µν φ (17)Assume the Lagrangian for the wave function is, L ( φ, D µ φ,g µν ) = D λ φ ∗ D λ φ − M φ ∗ φ (18)The Lagrangian implies the Euler-Lagrange equation which results in theKlein-Gordon equation. D λ D λ φ + M φ = 0 (19)The gauge transformation applied to the invariant Lagrangian defines thecurrent, j ν = ie (cid:0) φ † D ν φ − (cid:0) D ν φ † (cid:1) φ (cid:1) (20)The canonical energy-momentum tensor of the Klein-Gordon scalar field is, T νSµ = D µ φ † D ν φ + D µ φD ν φ † − δ νµ L (21)These are the subject of the “two divergences” referred to earlier.The divergence charged electric current j µ vanishes, and so satisfies thecontinuity equation.
14 15 . ∇ µ j µ = 0 (22)The divergence of the energy-momentum tensor T µν for matter equals theLorentz force,
16 17 18 hence does not vanish in general. ∇ ν ( T µν ) = F µν J ν (23)The familiar vector component notation for the electromagnetic field tensor F µν is F µν = − E x − E y − E z E x H z − H y E y − H z H x E z H y − H x (24)We pursue the non-abelian generalization of the two divergences, Eqs. (22)and (10) as an application, as well as motivation for the construction of a moregeneral gauge covariant derivative than Eq. (11). The exploration providedhere will not be specific as to the matrix representation, but instead try toexpose larger patterns applying to all representations.0 IV. THE LAGRANGIAN FOR A WAVE FUNCTION
Begin with the Lagrangian of a system evolving in space-time with matterrepresented as a wave function, as in Eq. (1). The space-time has a generalmetric g µν ( x ). Assume a scalar Lagrangian dependent upon the space-timecoordinates only through the wave function and fixed background metric, sothat there is no coordinate “ x ” appearing explicitly in the Lagrangian. L = L ( φ , D µ φ , g µν ) (25)The symbol “ D µ ” indicates a gauge covariant partial derivative with respectto coordinate x µ which, like all derivatives, satisfies the product rule (Leibnizrule). For now simply note that it does not commute as does the ordinarypartial derivative with respect to coordinates ∂/∂x µ , or in short form ∂ µ .The gauge covariant derivative of the metric is defined to vanish, the metricity condition. By definition the gauge covariant derivative D µ be-comes to the coordinate covariant derivative ∇ λ upon application to gaugeinvariant objects such as the metric. D λ g µν ( x ) = ∇ λ g µν ( x ) = 0 (26)As shown later, the Lagrangian defines equations of motion for the wavefunction through the application of the Euler-Lagrange equations which thencontrol the evolution of the wave equation. V. THE GAUGE TRANSFORMATION OF THE WAVE FUNCTION
Consider the following implementation of a gauge transformation as a ho-mogeneous nonsingular transformation applied to a column matrix φ wavefunction. Left-multiply the wave function with the complex-valued squarematrix T ( x ) which is then transformed to another column matrix, T φ , also a wave function. In what sense the transformed wave function is the sametype of wave function reaches deep into the meaning of this transformation.It will become clear that the transformation matrix must be drawn from themembers of a specific Lie group.Represent a gauge transformation with the arrow “ −→ g ” which means “re-place each occurrence” of the symbol on the left with the expression on theright. Another useful notation uses a “ c hat ” to indicate the gauge transformedexpression. With this notation, the gauge covariant transformation of the1wave function is, φ −→ g ˆ φ = T φ (27)A transformation may do nothing, in which case it is the identity trans-formation.Some objects are gauge invariant, such as the metric. g µν −→ g g µν (28)The gauge transformation may be regarded as part of the definition of thewave function, in a way that will be made clear.The gauge transformation is limited to unitary to conform to state repre-sentation in quantum mechanics, which is that of left and right vectors in acomplex space. The “covariant wave function” here is a “right vector”, and thecontravariant row matrix is the left vector. Inner products of a contravari-ant and covariant matrix (wave function) are defined to be gauge invariant. Acontravariant row matrix, left vector, wave function ψ is defined to transformas ψ −→ g ˆ ψ = ψ T − (29)Limit the gauge transformations to those of interest in quantum mechanics,where the norm of the wave function is required to be gauge invariant, φ † φ = X i φ ∗ i φ i −→ g d φ † φ = φ † φ (30)where φ † is the conjugate transpose of the wave function. The gauge con-travariant transformation required to assure invariance of the real-valued,positive definite norm is the conjugate transpose of Eq. (27), φ † −→ g c φ † = φ † T † (31)Substitute into Eq. (30). φ † φ −→ g φ † T † T φ (32)Invariance of the norm is achieved when the gauge transformation is limitedto unitary gauge transformations where, T † T = TT † = (33)2The gauge transformations examined here are unitary transformations. Un-der these transformations, any column matrix covariant wave function can be“converted” to a contravariant row matrix with the conjugate transpose oper-ation, to make an invariant inner product. The contravariant transformationis, ψ −→ g ˆ ψ = ψ T † (34)For completeness, the gauge transformation of the square matrix M isdefined so that the following product is invariant, φ † M φ −→ g φ † M φ = φ † T † ˆMT φ (35)which implies the transformation properties for the matrix , M −→ g ˆM = TMT † (36)We started with the definition of a covariant transformation in Eq. (27)as applied to a column matrix wave function, then continued to attach thespecific gauge transformation property to specific matrix shapes, • column, covariant, Eq. (27), • row, contravariant, Eq. (31), • square matrix, Eq. (36), • an invariant of any shape, Eq. (28).We will maintain this correspondence between the shape of the complex matrixand its gauge transformation properties.The wave function used here is a column matrix of complex-valued scalars,but several other models for wave functions are used in quantum mechanics.The Dirac equation for the spinor wave function of an electron may takento be classical Grassmann numbers which anticommute among themselves. The electron wave function becomes an operator in a number representationin its interactions with an electromagnetic field. Invariant Lagrangians are typically constructed as a sum of gauge invariant terms, and now we have the means to create such terms, such as the productbetween a contravariant and covariant wave function.3
VI. TRANSFORMATION PROPERTIES OF THE GAUGE COVARIANTDERIVATIVE
The gauge covariant derivative of the wave function is defined to transformthe same way as the wave function. Indicate the gauge transformation asapplied to the wave function and its derivative as, D µ φ −→ g [ D µ φ = T ( D µ φ ) (37)Put another way, the gauge covariant derivative does not change the transfor-mation properties of its operand, which in this case is the wave function. Thegauge transformations also applies to the operator form of the gauge covariantderivative D µ . D µ φ −→ g ˆ D µ ˆ φ = ˆ D µ T φ (38)Equate the two transformed derivatives on the right hand sides of Eqs (37)and (38). T D µ φ = ˆ D µ T φ (39)The wave function is arbitrary, and solve for the transformed gauge co-variant derivative operator using Eq. (33). D µ −→ g ˆ D µ = T D µ T † (40)This is the same gauge covariant property as a square matrix defined inEq. (36).The gauge covariant derivative is defined to reduce to the ordinary par-tial derivative when applied to an invariant scalar object. Apply the gaugecovariant derivative to an invariant such as the inner product between a con-travariant wave function ψ † , and a covariant wave function, φ , D µ (cid:0) ψ † φ (cid:1) = ∂ µ (cid:0) ψ † φ (cid:1) (41)Apply the product rule. D µ (cid:0) ψ † (cid:1) φ + ψ † D µ ( φ ) = ∂ µ (cid:0) ψ † φ (cid:1) (42)Create another expression by taking the conjugate transpose of the invariant,4again resulting in an invariant. Apply the gauge covariant derivative. D µ (cid:0) φ † ψ (cid:1) = ∂ µ (cid:0) φ † ψ (cid:1) (43)and the product rule again applies. D µ (cid:0) φ † (cid:1) ψ + φ † D µ ( ψ ) = ∂ µ (cid:0) φ † ψ (cid:1) (44)Take the conjugate transpose of Eq. (42). φ † D µ ( ψ ) + ( D µ ( φ )) † ψ = ∂ µ (cid:0) φ † ψ (cid:1) (45)Equate the left hand sides of Eqs. (44) and (45).( D µ ( φ )) † = D µ (cid:0) φ † (cid:1) (46)The conjugate transpose commutes with the gauge covariant derivative. Re-moving the arbitrary wave function, and again viewing the gauge covariantderivative as an operator, the operator must be Hermitian. This propertyarises from interpreting the conjugate transpose as having contravariant gaugetransformation properties. ( D µ ) † = D µ (47)Sufficient properties are defined here to begin constructing the gauge co-variant derivative. VII. CONSTRUCTION OF THE GAUGE COVARIANT DERIVATIVE
The definition of the gauge covariant derivative requires the introductionof an additional gauge field, A µ ( x ), also a square matrix of the same di-mension as the transformation matrix. Following the coordinate covariantderivative analogy, the gauge covariant derivative of the wave function φ isdefined the usual way by including gauge fields by the method of “minimumcoupling”.
27 28 29 30
Later we find that this field is an external field applyinga force to the wave function system. In the context of electromagnetics, thisnew field would be the magnetic potential A µ . Try the following form for thegauge covariant derivative operator as a generalization of the ordinary partialderivative.
32 33 D µ = ∂ µ − i A µ (48)5All parts of this Ansatz are invariant except for the unknown transformationproperties of the new field A µ . Substitute into Eq. (40). ∂ µ − i ˆA µ = T ( ∂ µ − i A µ ) T † (49)Solve for the transformed field. A µ −→ g ˆA µ = i T ∂ µ T † + TA µ T † (50)The new field is a square matrix, but compare to Eq. (36) to find an additionalterm which breaks the linear homogeneous transformation properties of asquare matrix.With the new field transforming as indicated, the gauge covariant deriva-tive defined in Eq. (49) satisfies the transformation requirements in Eq. (41).The gauge covariant derivative looks like the following when applied to thewave function. D µ φ = ∂ µ φ − i A µ φ (51)This form of the derivative satisfies the required transformation properties ofthe gauge covariant derivative.If there is a field we can add to A µ indicated by A ⊥ µ which commutes with the transformation matrix T , then Eq. (50) indicates( A µ + A ⊥ µ ) −→ g T ( A µ + A ⊥ µ ) T − + ( ∂ µ T ) T − (52)then A ⊥ µ is invariant under the gauge transformation. A ⊥ µ −→ g A ⊥ µ (53)Other commuting transformations do not enter the construction of the gaugecovariant derivative.The gauge covariant derivative may also be applied to a contravariant(row matrix) wave function ψ † . Substitute the construction into the productrule, Eq. (42), then solve for the derivative as applied to a contravariant wavefunction. D µ (cid:0) ψ † (cid:1) φ + ψ † ( ∂ µ φ − i A µ φ ) = ∂ µ (cid:0) ψ † φ (cid:1) (54)Eliminate the arbitrary covariant wave function with the result, D µ ψ † = ∂ µ ψ † + i ψ † A µ (55)6Further, the gauge covariant derivative may be applied to a square matrix.All that matters to the construction of the gauge covariant derivative is thegauge transformation properties of the operand, so create a square matrix byusing the outer product. D µ (cid:0) φψ † (cid:1) = φ∂ µ ψ † + i φψ † A µ + ( ∂ µ φ ) ψ † − i A µ φψ † (56)or D µ (cid:0) φψ † (cid:1) = ∂ µ (cid:0) φψ † (cid:1) + i (cid:2)(cid:0) φψ † (cid:1) , A µ (cid:3) (57)This can be generalized to an arbitrary square matrix M , D µ M = ∂ µ M + i [ M , A µ ] (58)which transforms as in Eq. (36).The gauge covariant derivative operator is Hermition as found in Eq. (47)which provides an additional property of the new field A µ .( ∂ µ − i A µ ) † = ∂ µ − i A µ (59)which implies that the new field is Hermitian. A µ = A † µ (60)Essential results come from introducing additional dependency among theelements of the matrix. To provide for further dependency, the transformationmatrix may be parameterized . VIII. PARAMETERIZATION OF THE TRANSFORMATION
It becomes extremely useful to assume that the transformation ma-trix T ( θ ) is a function of n independent, real-valued, parameters θ a , a =1 , , . . . n . The parameters are represented in the aggregate as θ . Each el-ement of the square transformation matrix is a function of the parameters.The parameterization of the transformation matrix is fixed for all calculationsto follow and are specific to a Lie group . Not all the matrix elements are in-dependent since the number of parameters is assumed fewer than the numberof elements in the matrix.Each of the infinite set of parameterized transformation matrices is uniquelydefined by the parameters, so that the parameters are the coordinates of the7transformation matrix T ( θ ) on the n dimensional manifold formed by theparameters. A specific transformation matrix T ( θ ) is located at θ on themanifold.Space-time dependency of the transformation matrix enters only throughthe parameters so that the full dependency is indicated by T ( θ ( x )). Positionin space-time does not explicitly appear in the transformation matrix. Space-time dependency of the transformation enters in a smooth differentiable wayonly through the parameters θ a ( x ).Parameterization of the transformation matrix provides unique identifica-tion of the matrices as elements of a Lie group with the addition of the groupaxioms. The group axioms each make physical sense as a model of gaugetransformations. For example, successive transformations of a wave functionagain transforms according to a transformation matrix from the same Liegroup.We proceed to the
Lie algebra , which is a linearization of the transfor-mation matrix near the identity element. However, little reference will bemade to group properties, since the principle of covariance provides sufficientguidance for the construction of the gauge covariant derivative.
IX. TAYLOR EXPANSION OF THE TRANSFORMATION
Expand the transformation matrix near the origin in a Taylor series. With-out loss of generality, assume that the identity resides at the origin of thecoordinates θ a = 0. T (0) = (61)Assume that the transformations are connected to the identity by a smoothdifferentiable path in parameter space. The continuity requirement allows usto take derivatives of the transformation matrix with respect to the parame-ters. The Taylor series is T = T (0) + ∂ T ( θ ) ∂θ a (cid:12)(cid:12)(cid:12)(cid:12) θ =0 θ a + O (cid:0) θ (cid:1) (62)Fix the point of evaluation at the origin, θ a = 0, then define the squarematrix constant generators t a where the imaginary i conveniently connects toa convention made apparent later. i t a = ∂ T ( θ ) ∂θ a (cid:12)(cid:12)(cid:12)(cid:12) θ =0 (63)8Approximate the transformation near the origin. Since the transformationmatrix must be close to the origin in this approximation, the start and theend points of the parameter difference δθ a must also be close to the origin. T ( δθ ) ≈ + i t a δθ a (64)The square matrix generators are summed via the repeating index a with asquare matrix result, t a δθ a = N X a =1 t a δθ a (65)Find that in order to limit our transformations to unitary in Eq. (33),( + i t a δθ a ) † ( + i t a δθ a ) = (66)the generators must be Hermitian. t † a = t a (67)The definition of the generators, Eq. (63) implies a number of properties.The generators have no parameter dependence, so by construction are gaugeinvariant. t a −→ g t a (68)Again by construction the generators are are not a function of position,therefore constant. ∂ µ t a = 0 (69)By definition the gauge covariant derivative of a gauge invariant (and coordi-nate invariant) object reduces to the ordinary partial derivative, so the partialderivative in Eq. (69) can be promoted to a gauge covariant derivative. D µ t a = 0 (70)The vanishing derivative of the generator has the additional advantage of pre-venting the generators from becoming dynamical objects, and it is an essentialproperty in the derivation of the charged current and its vanishing divergencewhich follows.It remains to define the construction of the gauge covariant derivative later.Much can be done using its properties without knowledge of its construction.9 X. REPRESENTATION OF THE GAUGE TRANSFORMATION MATRIX
The approximation to the gauge transformation matrix in Eq. (64) maybe repeated in a limiting process to calculate the matrix located at finiteparameter values, so extending the representation to finite distances from theorigin. T ( θ ) = lim k →∞ (cid:0) + i k t a δθ a (cid:1) k = ∞ X n =0 ( i t a θ a ) n n ! (71)or, using the exponentiation operator, T ( θ ) = EXP ( i t a θ a ) (72)This result is valid for real and complex numbers as well as square matrices.The transformation matrix yielded by this process is constrained by conti-nuity requirements for the Taylor series, hence this process may not yield thetransformation matrices in parts of the manifold not smoothly connected tothe origin. XI. INFINITESIMAL GAUGE TRANSFORMATION
Our calculations will use the properties of the wave function under in-finitesimal gauge variations, so we will explore the properties of the Lie alge-bra.The gauge transformation effectively adds parameter dependence to thetransformed wave equation ˆ φ ( θ, x ). Hence the partial derivative with respectto the parameter of the gauge transformed wave function is, ∂∂θ a ˆ φ = ∂∂θ a ( T ) φ (73)where the partial with respect to the parameter applies only to the transfor-mation matrix which has the only appearance of the parameter. The partialwith respect to the parameter of the wave function and evaluated at the originis now, ∂∂θ a ˆ φ (cid:12)(cid:12)(cid:12) θ =0 = i t a φ (74)Motivated by use of the partial derivative in Eq. (73), define a differenceor variation operator δ which is in turn defined by the infinitesimal gaugetransformation. The variation operator applies to the wave function just asthe partial derivative, but with the evaluation at parameter zero “built in” to0the notation. (cid:0) δδθ a (cid:1) φ ≡ ∂∂θ a ˆ φ (cid:12)(cid:12)(cid:12) θ =0 = i t a φ (75)Substitute Eq. (64) into the gauge transformation of the wave functionEq. (37), so that for parameter values infinitesimally near the origin, δθ a ,defines a change indicated by δ ,ˆ φ = φ + δ φ = ( + i t a δθ a ) φ (76)so that the infinitesimal gauge transformation is, δ φ = ( i t a φ ) δθ a δ ( D µ φ ) = ( i t a D µ φ ) δθ a (77)or in the form of a derivative, (cid:0) δδθ a (cid:1) φ = i t a φ (cid:0) δδθ a (cid:1) ( D µ φ ) = i t a ( D µ φ ) (78)Contravariant objects have the transformation property, δ φ † = (cid:0) − i φ † t a (cid:1) δθ a δ (cid:0) D µ φ † (cid:1) = (cid:0) − i (cid:0) D µ φ † (cid:1) t a (cid:1) δθ a (79)Substitute Eq. (64) into Eq. (36) to find the infinitesimal gauge transformationfor a square matrix. δ M = i [ t a , M] δθ a (80)Again, the generators are gauge invariant, according to Eq. (68). δ t a = 0 (81)and the metric is gauge invariant, δg µν = 0 (82)The infinitesimal gauge transformation of A µ follows from substitutingEq. (64) into the transformation Eq. (50). ˆA µ = i ( + i t a δθ a ) ∂ µ (cid:0) − i t b δθ b (cid:1) + ( + i t a δθ a ) A µ (cid:0) − i t b δθ b (cid:1) (83)1so that, ˆA µ = t b ∂ µ δθ b + A µ + i [ t a , A µ ] δθ a (84)or δ A µ = t b ∂ µ δθ b + i [ t a , A µ ] δθ a (85)This completes the application of the infinitesimal gauge transformation tothe wave function, and its derivative, and generators of the gauge transforma-tion. However, the toolkit required to construct gauge invariant Lagrangiansis not complete, since terms in the Lagrangian may include the parameterindex. The gauge transformation properties of column, row and square ma-trices are defined above, but remaining to be defined is gauge transformationproperties of the parameter index of the generators. The definition of gaugeinvariance of the generators will lead to Sophus Lie’s second theorem, andconsistent infinitesimal transformation properties of the parameter index. XII. GAUGE INVARIANT GENERATORS, SOPHUS LIE’S SECONDTHEOREM
The generators are defined to be gauge invariant, then the statement ofthat invariance, Eq. (81) links two infinitesimal homogeneous gauge transfor-mations, one for the matrix indices as already discussed, and the other forthe parameter index, which has not been examined. Exactly analogous is thesimultaneous gauge transformation of the wave function and electromagneticfield as outlined in Eqs. (12) and (13).Each generator is a square matrix so that the matrix part of the infinites-imal gauge transformation must look like Eq. (80). Now we turn attention tohow a parameter index is transformed under a gauge transformation.The generators t a form a basis for a vector space. One is then free toredefine the generators by a real nonsingular linear transformation with a cor-responding redefinition of the parameters . The required invariance of thesegenerators as stated in Eq. (81) dictates that the parameters must transformin such a way as to preserve invariance under infinitesimal variations of theparameters. These infinitesimal variations, because of their smallness, maybe expected to effect a homogeneous transformation of the generators.We have defined the infinitesimal gauge transformation for column, rowand square matrices. Now to find the infinitesimal gauge transformation of the parameter index. With the left and right multiplications of wave functions,the only free index in the expression ψ † t a φ is the covariant parameter index“ a ”. This parameter index must transform under a gauge transformation in2order to satisfy the requirement that the generators are gauge invariant.In the same spirit as the homogeneous transformation of the wave func-tion in Eq. (27), we define the most general homogeneous transformation interms of the set of constants f cba which will turn out to be the structure con-stants of the Lie algebra . Use the structure constants to define the followingtransformation for the covariant (lower) parameter index. (cid:0) δ/δθ b (cid:1) (cid:0) ψ † t a φ (cid:1) = f cba (cid:0) ψ † t c φ (cid:1) (86)Generalize this infinitesimal transformation property to any lowered parame-ter index such as an arbitrary complex-valued vector V a ( θ ). (cid:0) δδθ b (cid:1) V a = f cba V c (87)An inner product with a contravariant parameter vector W a is invariant. δ ( V a W a ) =0 (88)Once again, this invariance requirement leads to the infinitesimal gauge trans-formation for a contravariant parameter vector (raised index). (cid:0) δδθ b (cid:1) V a = − f abc V c (89)A procedure exists to convert a covariant to a contravariant vector via the Cartan-Killing metric which will be described shortly. The procedure is anal-ogous to raising and lowering coordinate tensor indices with the metric tensor,and will be defined shortly.We have sufficient definitions to elaborate the infinitesimal gauge invari-ance of the generators as stated in Eq. (68). The requirement for invari-ant generators links the two homogeneous gauge transformations, one for thesquare matrix generators, Eq. (80), and the other for the lowered parameterindex Eq. (87). δδθ b ( t a ) = f cba t c + i [ t b , t a ] ≡ [ t a , t b ] = if cab t c (91)where the structure constants are defined to be antisymmetric. f cab = − f cba (92)Eq. 92) is Lie’s Second Theorem (Marius Sophus Lie 1842 1899), whichwe will refer to as closure under the operation of commutation. This the-orem follows from the requirement that both matrix and parameter indicestransform homogeneously in such a way as to preserve the infinitesimal gaugeinvariance of the generators. As pointed out in the references, imposition ofgroup axioms on the gauge transformation matrix leads to the same destina-tion.
XIII. VANISHING DERIVATIVE OF THE GENERATORS
The gauge covariant derivative of the generators vanish in Eq. (70). Thismust be confirmed with the explicit definition of the derivative in Eq. (51).The gauge covariant derivative must be extended to include parameter indicesas well as matrix indices. Exploit the deep analogy to the coordinate co-variant derivative by adding a term, Γ cµa , analogous to the Christoffel symbolappearing in the coordinate covariant derivative. D µ t a = ∂ µ t a + Γ cµa t c − i [ A µ , t a ] = 0 (93) Assume that the additional field, A µ ( x ) is within the span of the generatorsused as basis functions.
44 45 A µ = A aµ t a (94)where the coefficients A aµ ( x ) of the basis functions carry the position depen-dency. These coefficients are the multicomponented gauge potential . Since A µ is Hermitian, Eq. ( ?? ), and the generators are Hermitian, then the gaugepotential is real-valued which fits the convention for the electromagnetic 4-potential. (cid:0) A aµ (cid:1) ∗ = A aµ (95)Substitute Eqs. (69), and (94) into Eq. (93).Γ cµa t c − i (cid:2) A bµ t b , t a (cid:3) = 0 (96)4Substitute Lie’s Second Theorem, Eq. (91).Γ cµa t c = A bµ f cab t c (97)Remove the generator dependency.Γ cµa = A bµ f cab (98)The vanishing covariant derivative of the generators provides a consistentcalculation, Eq. (94) for the additional field, A µ ( x ). Substitute Eqs. (94) and(98) into (93). D µ t a = ∂ µ t a + A bµ f cab t c − iA bµ [ t b , t a ] = 0 (99)Insert gauge potentials via Eq. (94) into the gauge covariant derivative,Eqs. (51), (55) and (58). D µ φ = ∂ µ φ − iA aµ t a φ (100) D µ ψ † = ∂ µ ψ † + iA aµ ψ † t a (101) D µ M = ∂ µ M + iA aµ [ M , t a ] (102)The gauge transformation of the gauge potentials A aµ ( x ) follows from sub-stituting Eq. (94) into Eq. (85). δ (cid:0) A bµ t b (cid:1) = t b ∂ µ δθ b + i (cid:2) t a , A bµ t b (cid:3) δθ a (103)Extract the gauge potential from the commutator, note that the generatorsare gauge invariant, then substitute Lie’s Second Theorem, Eq. (91). δ (cid:0) A bµ (cid:1) t b = t b ∂ µ δθ b − A cµ f bac t b δθ a (104)Remove the generator dependency, and rename indices. δA aµ = ∂ µ ( δθ a ) − A cµ f abc δθ b (105)Interestingly, this relation has no matrix indices, so that it is independent ofthe matrix representation of the generators.5 XIV. VECTORS IN PARAMETER SPACE
Generalize the gauge covariant derivative to an arbitrary covariant vector V a ( x ), and contravariant vector V a ( x ) with a parameter index. Define thefollowing, consistent with Eq. (99), D µ ( V a ) = ∂ µ ( V a ) + A bµ f cab V c D µ ( V a ) = ∂ µ ( V a ) − A bµ f acb V c (106)so that the derivative of an invariant becomes the ordinary partial. D µ ( V a V a ) = ∂ µ ( V a V a ) (107)The gauge covariant derivative and gauge transformation can now be consis-tently applied to arbitrary contravariant wave functions and vectors in pa-rameter space.The gauge potential transformation in Eq. (105) can be seen to beclosely related to contravariant parameter transformation when compared toEq. (106). In fact, the gauge transformation can be written in an interestingform in terms of a derivative. δA aµ = D µ ( δθ a ) (108)The form of the additional field in Eq. (94) may be generalized by adding aterm which commutes with all the generators. The derivative of the generatorsstill vanish with this term, so this is a consistent modification, the implicationof which is not pursued here. XV. DEFINITION OF THE FIELD STRENGTH TENSOR
Needed shortly is the commutator of the gauge covariant derivative whichcan be evaluated given the definition of the derivative, Eq. (100), constantgenerators, and Lie’s second theorem Eq. (91).[ D µ , D ν ] φ = iF aµν t a φ (109)where the definition of the field strength tensor F aµν for the gauge potentials A aµ is F aµν = − ∂ µ A aν + ∂ ν A aµ − A bµ A cν f abc (110)6The field strength tensor is free of any matrix indices, hence is independent ofthe matrix representation of the generators. As indicated by the contravariantparameter index the field strength tensor is not gauge invariant, (cid:0) δδθ b (cid:1) F aµν = − f abc F cµν (111)whereas the electromagnetic field tensor is gauge invariant.This completes the set of tools required to do gauge covariant calculations.All objects such as generators, wave functions, field strength tensor, currentvectors, have well defined transformation properties and derivatives. XVI. DEFINITION OF THE CARTAN-KILLING METRIC
We use the Cartan-Killing metric g ab to raise and lower parameter indices,so “converting” one to the other.Apply the CartanKilling inner product which is defined as the trace of thematrix product of two generators. The curly brackets indicate applicationof the matrix Trace operation. g ab ≡ T r { t a t b } (112)By this definition, the metric is symmetric since matrices commute under thetrace. g ab = g ba (113)The metric, constructed from gauge invariant generators, is gauge invari-ant. δδθ c ( g ab ) = 0 (114)Substitute the variation for a covariant parameter vector, Eq. (87). δδθ c ( g ab ) = f eca g eb + f ecb g ae = 0 (115)Define the covariant structure constants, f abc = f dbc g da (116)then, δδθ c ( g ab ) = f bca + f acb = 0 (117)which shows that the covariant structure constants f bca are antisymmetric in7the ab indices. f bca = − f acb (118)The definition of the covariant structure constants implies antisymmetry inthe ca indices, so that they are completely antisymmetric.Similarly, we expect the gauge covariant derivative of a function of thegenerators to vanish. D µ ( g ab ) = ∂ µ g ab + A dµ f cad g cb + A dµ f cbd g ac = 0 (119)The CartanKilling metric is constant, and the external gauge potential arbi-trary, so that the derivative vanishes due to the antisymmetry of the covariantstructure constants.Indicate the inverse of the Cartan-Killing metric as g ab where g ac g bc = δ ab (120)The metric and its inverse can be used to “convert” parameter indices, sothat by definition, given a contravariant vector V a the corresponding covariantvector is V b = g ab V a (121)The contravariant parameter index field tensor F aµν is defined in Eq. (110).Construct the covariant parameter field tensor using the Cartan-Killing met-ric, F a µν = g ab F aµν (122)Construct a gauge and coordinate invariant Lagrangian analogous to electro-magnetics, where contravariant parameter indices must be summed against acovariant index such as L gauge (cid:0) F aµν (cid:1) = F µνa F aµν (123)Compare this to the electromagnetic Lagrangian, also gauge and coordinateinvariant. L elect ( F κη ) = F κη F κη (124)The problem of constructing contravariant parameter vectors uses theCartan-Killing Lie algebra metric which may be used to raise and lower pa-rameter indices, analogous to the coordinate metric.8 XVII. INVARIANT STRUCTURE CONSTANTS, OR THE JACOBIIDENTITY
We briefly continue our foray into Lie algebra, in the guise of the infinitesi-mal gauge transformation. Quadratic constraints upon the structure constants f c ab follow from the Jacobi identity. Evaluate the
Jacobi identity which isthe commutator of the matrix generators summed with permutations of theindices. [[ t a , t b ] , t c ] + [[ t b , t c ] , t a ] + [[ t c , t a ] , t b ] = 0 (125)This expression is identically zero with expansion of the commutators, andapplication of the associative property for matrix operations. The Jacobiidentity becomes a consistency constraint on the structure constants by re-peatedly substituting Eq. (91). f dab f edc t e + f dbc f eda t e + f dca f edb t e = 0 (126)Without constraining the generators, the structure constants must satisfy thequadratic constraint, f dab f edc + f dbc f eda + f dca f edb = 0 (127)Note that the quadratic constraint implies gauge invariance of the structureconstants via Eqs. (87) and (89). (cid:0) δ/δθ f (cid:1) ( f eab ) = f dab f edc + f dbc f eda + f dca f edb = 0 (128)View the Jacobi identity as a consistency requirement since it implies invari-ance of the structure constants, which is also consistent with invariance of thegenerators. XVIII. THE GAUGE TRANSFORMATION OF THE LAGRANGIAN
Now consider the two “divergence” applications of the gauge covariantderivative mentioned earlier. Both involve the Lagrangian. Consider thegauge transformation of the Lagrangian. L ( φ , D µ φ , g µν ) −→ g ˆ L = ˆ L ( T φ , T D µ φ , g µν ) (129)9where ˆ L is the gauge transformed Lagrangian with the replacement indicatedin Eqs. (27), and (37). The transformation adds the matrix at each appearanceof the wave function and its derivative. The replacement adds a parameterdependency to the Lagrangian. Evaluated at parameter zero, the value ofthe Lagrangian is unchanged since the transformation matrix reduces to theidentity matrix.ˆ L ( T ( θ ) φ , T ( θ ) D µ φ , g µν ) (cid:12)(cid:12)(cid:12) θ =0 = L ( φ , D µ φ , g µν ) (130)Apply the partial derivative with respect to θ to ˆ L . Since the addition ofthe transformation matrix does not change the functional dependence, we canevaluate the variation using the chain rule with respect to the wave functionand its derivative. (cid:0) δδθ a (cid:1) L = ∂L∂ φ (cid:0) δδθ a φ (cid:1) + ∂L∂D µ φ (cid:0) δδθ a D µ φ (cid:1) (131)Substitute the variations, Eq. (78). (cid:0) δδθ a (cid:1) L = ∂L∂ φ i t a φ + ∂L∂D µ φ i t a D µ φ (132)This the starting point for determining the divergence of the current as definedby an infinitesimal gauge transformation. XIX. THE DIVERGENCE OF THE CHARGED CURRENT
The vanishing divergence of the current will be our first “vanishing diver-gence” relation. The second vanishing divergence soon to follow applies to theenergy-momentum tensor.The constant generators expressed in Eq. (70) implies that the generatorcommutes with the derivative. t a D µ φ = D µ ( t a φ ) (133)so that the variation of the Lagrangian Eq. (132) becomes, (cid:0) δδθ a (cid:1) L = ∂L∂ φ i t a φ + ∂L∂D µ φ D µ ( i t a φ ) (134)0Rearrange with application of the product rule, (cid:0) δδθ a (cid:1) L = (cid:26) ∂L∂ φ − D µ (cid:18) ∂L∂D µ φ (cid:19)(cid:27) t a φ + D µ (cid:18) ∂L∂D µ φ t a φ (cid:19) (135)Identify the equations of motion for the wave function φ in the curly brackets.The row-matrix expression Λ is set to zero to arrive at the Euler-Lagrangeequation Λ = 0. Λ = ∂L∂ φ − D µ (cid:18) ∂L∂D µ φ (cid:19) = 0 (136)Note: The many examples of wave equations defined by the Euler-Lagrange equation applied to an appropriate Lagrangian include elec-tromagnetics, the Klein-Gordon wave equation for spin zero fields andthe Dirac wave equation for spin one-half fields. Satisfaction of theEuler-Lagrange equation, Λ = 0, defines the term “on shell”. Offshell dynamics violate the Euler-Lagrange equation.For a gauge invariant scalar Lagrangian, the partial with respect to thecovariant wave function yields a contravariant result. Similarly, constructionof Λ which includes partials with respect to covariant quantities, indicatesthat it transforms contravariantly, Eq. (31). Λ −→ g ˆ Λ = ΛT † (137)Upon setting the contravariant Euler-Lagrange equations to zero as an equa-tion of motion, subsequent gauge transformations will maintain the zero valuefor ˆ Λ , and so maintain the equations of motion. Each parameter of the gauge transformation has a corresponding currentas indicated by the covariant (lower) index “ a ” in the definition , J µa = ∂L∂D µ φ t a φ (138)Substitute the Euler-Lagrange expression Eq. (136) and the current intoEq. (135), (cid:0) δδθ a (cid:1) L = Λt a φ + D µ J µa (139)Solve for the divergence of the gauge current. D µ J µa = (cid:0) δδθ a (cid:1) L − i Λt a φ (140)1The gauge current has a vanishing divergence with satisfaction of 1) the Euler-Lagrange equation, and 2) a symmetry of the Lagrangian which is defined as, (cid:0) δδθ a (cid:1) L = 0 (141)A gauge transformation which is also a symmetry of the Lagrangian, leaves theLagrangian invariant. The divergence of the current in Eq. (140) may be re-garded as an identity since identity since it is dependent upon only definitionsof quantities, and the product rule.The current is defined in Eq. (138) whether or not the gauge transformationis a symmetry of the Lagrangian.The definition of a charged current, and the conditions for the vanishingdivergence of the current is central in physics, and, for electromagnetics, im-plies charged current continuity. This relationship motivates and rewards ourpursuit for a gauge covariant derivative that commutes with the generators,Eq. (133). XX. THE DIVERGENCE OF THE ENERGY-MOMENTUM TENSOR
The energy-momentum of a system responds to external forces acting uponthe system according to a “divergence law” as applied to energy-momentum.These interesting results can be found without detailing the construction ofthe gauge covariant derivative. All that is required is the chain rule, productrule, and the non-commuting gauge covariant derivative.Apply the gauge covariant derivative to the scalar Lagrangian, Eq. (25),and use the chain rule. D µ L = ∂L∂ φ D µ φ + ∂L∂ ( D ν φ ) D µ ( D ν φ ) + ∂L∂ ( g λν ) D µ g λν (142)The last term vanishes by the metricity condition above, Eq. (26). Rearrange-ment using only the product rule results in D ν (cid:26) Lδ νµ − ∂L∂D ν φ D µ φ (cid:27) = ∂L∂D ν φ [ D µ , D ν ] φ + (cid:26) ∂L∂ φ − D ν (cid:18) ∂L∂D ν φ (cid:19)(cid:27) D µ φ (143)Verify this expression by expanding the curly brackets, and finding that the2end result is Eq. (142), the chain rule. Conversely, prove this expression bynoting that each step involving the product rule is reversible.Identify the equations of motion for the wave function φ provided by therow-matrix Euler-Lagrange equation Λ = 0.Identify in Eq. (143) the canonical energy-momentum tensor (EMT) T µν ,which is the covariant generalization of energy and momentum in space-time, T µν = Lδ νµ − ∂L∂D ν φ D µ φ (144)The canonical energy-momentum is the covariant generalization of energyand momentum in space-time. Substitute the EMT and the Euler-Lagrangeexpression Λ into Eq. (143) to find the divergence of the EMT. The gaugecovariant derivative D ν becomes the ordinary covariant derivative ∇ ν whenapplied to the gauge-invariant tensor EMT. ∇ ν ( T µν ) = (cid:18) ∂L∂D ν φ (cid:19) [ D µ , D ν ] φ + Λ D µ φ (145)The divergence of the canonical energy-momentum tensor (EMT), immedi-ately relates to the Euler-Lagrange equation of motion, and a commutator ofthe gauge covariant derivative. This relation is a consequence of 1) the form ofthe Lagrangian, 2) the chain rule and the product rule. The non-commutinggauge covariant derivative must be defined as applied to the wave function,and it remains to associate this term with a force.The divergence of the EMT vanishes when the commutator vanishes (noexternal forces), and the Euler Lagrange equations are satisfied. In the spe-cial case of electromagnetics acting on the wave function in flat space, this“vanishing divergence” represents conservation of energy and momentum. Note the that this relation can be applied to any function of componentswhich supports a derivative. The Lagrangian is not required to be a scalar,nor the indices tensor indices. The relation is an identity with the definitionsand the functional dependency of the Lagrangian. Mathematically, there isno requirement to be related to Physics!However, Physics is our domain, and ours to discover the mathematicswhich allows interpreting the term with the commutator as a generalizedLorentz force which is applied externally.3
XXI. THE GENERALIZED LORENTZ FORCE
We now have the definitions available to show that the electromagneticfield tensor acts on the current to subject the system to the Lorentz force.Substitute the result of the commutation Eq. (109) into the identity Eq. (145), D ν ( T µν ) = i ∂L∂D ν φ F µν φ + Λ D µ φ (146)Substitute the current defined in Eq. (138). D ν ( T µν ) = F µν j ν + Λ D µ φ (147)This is the final form of the identity between the divergence of the energy mo-mentum tensor, the generalized Lorentz force, and the Euler-Lagrange equa-tion. In this form, it becomes clear that the external field F µν acts on thecurrent j µ , and so exchanging energy with the system as determined by thedivergence of the energy momentum tensor.It should be noted that the Lorentz force relation, Eq. (147) has beenderived without assuming any space-time homogeneity, or symmetry of theLagrangian.The generalized Lorentz force in Eq. (147) is a generalization of the elec-tromagnetic Lorentz force equation in Eq. (23). XXII. VANISHING DIVERGENCE MAY NOT IMPLY LOCALCONTINUITY
In electromagnetics, the vanishing divergence implies continuity equationsfor the components of charged current. However, for non-abelian gauge trans-formations, a vanishing divergence of the current no longer immediately im-plies continuity. The additional gauge covariance indicated by the parameterindex “ a ” in the current J µa . The divergence of the current, Eq. (140), adds aterm including the externally applied potential. D µ J µa = ∇ µ J µa + A bµ f cab J µc = 0 (148)where the tensor covariant derivative is represented by ∇ µ . The additionalterm involving the gauge potential prevents the vanishing gauge covariantdivergence from becoming a continuity equation.The covariant divergence is brought into a form exposing ordinary partial4derivatives by using an identity for the covariant derivative, Eq. (10). D µ J µa = 1 √ g ∂ µ ( √ gj µ ) + A bµ f cab J µc = 0 (149)This form exposes the partial derivatives in the context of curved space, indi-cating explicitly that the divergence of partial derivatives fails to vanish dueto the gauge potential term, hence continuity fails.Continuity also fails when applied to a second order tensor such as theEMT, because of the additional term involving metric derivatives. ∇ ν ( T µν ) = 1 √− g ∂ ( T µν √− g ) ∂x ν − T νβ ∂g βµ ∂x ν = 0 (150)Vanishing divergence equations fail to represent differential continuityequations in two important cases, energy-momentum in curved space andnon-Abelian gauge fields. XXIII. CONCLUSION
Two “divergence-type” relations, one for the charged current vector, andthe other for the energy-momentum tensor motivate the Lie algebra machineryrequired to construct a gauge covariant derivative of a wave function. Thegauge covariant derivative is essential to the definition of charged currents,and their vanishing divergence. The gauge covariant derivative is a means forincorporating an external generalized Lorentz force, and, when this force iszero, the vanishing divergence of energy-momentum tensor.The definition of the charged current arises from the definition of thegauge transformation, whether or not it is a symmetry of the Lagrangian.The charged current is acted upon by the generalized Lorentz force due to theexternal field, exchanging energy with the system according to a divergencerelation for the energy momentum tensor of the system.Although the principle of least action is not used here, the two divergencerelations provide sufficient motivation for the Euler-Lagrange equations asthe equation of motion defined by a Lagrangian. It is interesting that thisapproach allows consideration of “off-shell” equations of motion.The initial assumption of covariance of the wave function is sufficient todiscover all transformation properties of the wave function and its gauge co-variant derivative. Further, the assumption of covariance guides the construc-5tion of the gauge covariant derivative. The types of covariance considered:covariant, contravariant, invariant, and gauge field transformation. Of these,only the infinitesimal transformation of the gauge field is not a homogeneouslinear transformation.Lie’s Second Theorem follows from the principle of covariance which ap-pears as the requirement that both matrix and parameter indices transformhomogeneously in such a way as to preserve the infinitesimal gauge invarianceof the generators.The Lagrangian is assumed to not have an explicit appearance of the ex-ternal gauge field, since it appears only within the gauge covariant derivative.Continuity implied by vanishing “divergence-type” equations is lost whengeneralizing the partial derivative to the gauge covariant derivative becauseof an additional term not generally vanishing.
ACKNOWLEDGMENTS
I thank Jacques Rutschmann for providing just the right amount of skep-ticism and encouragement.
Appendix A: Example: the wave equation in a gauge field
Consider an example of the Klein-Gordon wave equation for a complexscalar field φ interacting with an external electromagnetic field which appearsonly in the covariant derivative. As is shown in many places, the Lagrangianwhich yields the Klein-Gordon wave equation of motion, Eq. (19) is, L ( φ, D µ φ,g µν ) = D λ φ ∗ D λ φ − M φ ∗ φ (A1)However, here we examine the Lagrangian for a wave function consisting ofa column matrix of scalars, represented as φ which will yield a wave equationsimilar to the Klein-Gordon. The following is independent of the specificvalue of the structure constants which may be substituted later. Again, theLagrangian has a similar appearance to the Lagrangian for the Klein-Gordonequation. L ( φ , D µ φ ,g µν ) = D λ φ † D λ φ − M φ † φ (A2)Apply the gauge covariant derivative to the Lagrangian, and use the chainrule, following the same steps as in the derivation for the energy momentum6tensor Eq. (142) then followed by its divergence Eq. (147). D µ L = D µ (cid:0) D λ φ † (cid:1) D λ φ + D λ φ † D µ ( D λ φ ) − M D µ (cid:0) φ † (cid:1) φ − M φ † D µ φ (A3)Add and subtract the second order derivative, but with swapped indices. Usea commutator bracket, [ D µ , D λ ] φ = D µ D λ φ − D λ D µ φ (A4)so that, D µ L = (cid:0) [ D µ , D λ ] φ † (cid:1) D λ φ + D λ (cid:0) D µ φ † (cid:1) D λ φ + D λ φ † ([ D µ , D λ ] φ ) + D λ φ † D λ ( D µ φ ) − M D µ (cid:0) φ † (cid:1) φ − M φ † D µ φ (A5)Apply the product rule, and rearrange to expose the Euler-Lagrange equationsof motion which are the coefficient of the derivative of the field D λ φ . D µ L = D λ (cid:0) D µ φ † D λ φ (cid:1) − (cid:0) D µ φ † (cid:1) (cid:0) D λ D λ φ + M φ (cid:1) + D λ (cid:0) D λ φ † D µ φ (cid:1) − (cid:0) D λ D λ φ † + M φ † (cid:1) ( D µ φ )+ (cid:0) [ D µ , D λ ] φ † (cid:1) D λ φ + D λ φ † ([ D µ , D λ ] φ ) (A6)Identify the Euler-Lagrange equation, and its complex conjugate transpose.These are the generalized Klein-Gordon equation. D λ D λ φ + M φ = 0 D λ D λ φ † + M φ † = 0 (A7)Substitute these equations, and collect terms under the derivative. D µ L = D λ (cid:0) D µ φ † D λ φ + † D µ φ D λ φ (cid:1) + (cid:0) [ D µ , D λ ] φ † (cid:1) D λ φ + D λ φ † ([ D µ , D λ ] φ ) (A8)Insert the Kronecker delta, move terms to the left hand side. Rename dummyindices. D ν (cid:2) D µ φ † D ν φ + D ν φ † D µ φ − δ νµ L (cid:3) = − (cid:0) [ D µ , D ν ] φ † (cid:1) D ν φ − D ν φ † ([ D µ , D ν ] φ ) (A9)The energy-momentum tensor T Sµν for the wave function appears inside7the curly bracket. T νSµ = D µ φ † D ν φ + D ν φ † D µ φ − δ νµ L (A10)Substitute the commutator results, Eq. (109) into Eq. (A9). D ν T νSµ = iF aµν (cid:0) φ † t a D ν φ − (cid:0) D ν φ † (cid:1) t a φ (cid:1) (A11)The charged current is defined as, j νa = i (cid:0) φ † t a D ν φ − (cid:0) D ν φ † (cid:1) t a φ (cid:1) (A12)then recognize the Lorentz force on the right hand side of the following, D ν T νSµ = F aµν j νa (A13)The divergence of the canonical energy-momentum tensor is equal to the gen-eralized Lorentz force, assuming satisfaction of the Euler-Lagrange equation.As expected, application of a generalized Lorentz force to the charged currentresults in an exchange of energy with the system.Consider the Abelian case where the generators all commute with eachother so that the structure constants are zero. Mutual commutation allowsthe generators to be simultaneously diagonalized. We will discover in thisinteresting case, that the model “splits up” and becomes a set of independentcomplex fields, each obeying the Klein-Gordon equation.The column matrix form of the wave function may be made explicit byindexing the components so that φ ( i ) represents a single complex value, and( i ) indicates a matrix index, and not a parameter index. φ ( x ) = φ (1) ( x )... φ ( n ) ( x ) (A14)then substituting into the Lagrangian for the scalar field, Eq. (A2), findthat the Lagrangian becomes a sum of individual Lagrangians. L = X i L ( i ) (A15)8Examine how this “splitting” comes about. The φ † φ term is φ † φ = X i φ ∗ ( i ) φ ( i ) (A16)The gauge covariant derivative is diagonal since each generator is diagonal. D ( i ) µ φ ( i ) = ∂ µ φ ( i ) − iA aµ t ( ii ) a φ ( i ) (A17)so that the “kinetic energy” term becomes, D λ φ † D λ φ = X i D ( i ) µ φ ∗ ( i ) D ( i ) µ φ ( i ) (A18)The gauge covariant derivative does not mix components so that each La-grangian is independent. L ( i ) = D ( i ) µ φ ∗ ( i ) D ( i ) µ φ ( i ) − M φ ∗ ( i ) φ ( i ) (A19)Each component satisfies the Euler-Lagrange equation.Λ (cid:0) L ( i ) (cid:1) = D λ ( i ) D ( i ) λ φ ( i ) + M φ ( i ) = 0 (A20)Similarly, the quadratic form of the current j νa in Eq. (A12), splits into asum of components. j νa = X i j νa ( i ) (A21)where j νa ( i ) = i (cid:16) φ ∗ ( i ) t ( ii ) a D ν ( i ) φ ( i ) − (cid:16) D ν ( i ) φ ∗ ( i ) (cid:17) t ( ii ) a φ ( i ) (cid:17) (A22)and again for the canonical energy momentum tensor, T νµ = X i T ν ( i ) µ (A23)We see that each complex scalar component of the column matrix of the field φ “feels” the same external electromagnetic field, and evolves independentlyof the other components of the field, without interaction between the scalarfield components. D ν T ν ( i ) µ = F aµν j νa ( i ) (A24)9An interesting task would be to extend the model, including interactionsbetween the field components.The charged current is identified, but not yet determined to have a van-ishing divergence which connects to a symmetry of the Lagrangian. Appendix B: Example: the current
The current is defined by the infinitesimal gauge transformation Eq. (77)and realized in Eq. (138). Apply the infinitesimal gauge transformation tothe Lagrangian in Eq. (A2), and use the chain rule, following the same stepsas in the derivation for the current in then followed by the divergence of thecurrent Eq. (140). δL = δ (cid:0) D λ φ † (cid:1) D λ φ + D λ φ † δ ( D λ φ ) − M δ φ † φ − M φ † δ φ (B1)Upon applying the infinitesimal gauge transformation and its complex conju-gate, it is immediately apparent that the Lagrangian is gauge invariant sincethe terms cancel. (cid:0) δδθ a (cid:1) L = 0 (B2)Current conservation may be derived from this, but only if we are not so quickto cancel terms. Substitute the variations, Eq. (78), and do not cancel terms. (cid:0) δδθ a (cid:1) L = − iD λ φ † t a D λ φ + iD λ φ † t a D λ φ + iM φ † t a φ − iM φ † t a φ (B3)It might seem strange working with terms which sum to zero, but that isexactly the step performed for the general Lagrangian. In each of the twoterms containing derivative, create a divergence, then subtracting the extraterm required by the product rule. (cid:0) δδθ a (cid:1) L = − iD λ (cid:0) φ † t a D λ φ (cid:1) + i φ † t a (cid:0) D λ D λ φ (cid:1) + iD λ (cid:0) D λ φ † t a φ (cid:1) − i (cid:0) D λ D λ φ † (cid:1) t a φ + iM φ † t a φ − iM φ † t a φ (B4)0Rearrange to expose the Euler-Lagrange expression, Eq. (A7) for this La-grangian. (cid:0) δδθ a (cid:1) L = − iD λ (cid:0) φ † t a D λ φ (cid:1) + i φ † t a (cid:0) D λ D λ φ + M t a φ (cid:1) + iD λ (cid:0) D λ φ † t a φ (cid:1) − i (cid:0) D λ D λ φ † + M φ † (cid:1) t a φ (B5)and the remaining terms are the divergence of the gauge invariant current. (cid:0) δδθ a (cid:1) L = − iD λ (cid:0) φ † t a D λ φ − (cid:0) D λ φ † (cid:1) t a φ (cid:1) (B6)so that the current, Eq. (A12), is conserved, if the infinitesimal gauge trans-formation represents a symmetry. (cid:0) δδθ a (cid:1) L = − D λ j λa = 0 (B7)This calculation derives from a Lagrangian the equation of motion forcharged matter, represented by the complex wave function, in an external field,Eq. (A7). The divergence of the canonical energy momentum tensor equalsthe Lorentz force as it acts on a current which has a vanishing divergence, asit must, in order to be identified as the charged current interacting with theexternal gauge field tensor. Appendix C: Example: Yang-Mills equations of motion
The Lagrangian for the electromagnetic field L em ( F κη ) = F κη F κη (C1)is a quadratic in field strength. The generalization of the Lagrangian fora gauge field tensor may be taken to be L gauge ( F κηa ) = F κηa F aκη (C2)where the parameter index is raised and lowered as needed with the Cartan-Killing metric g ab . We will discover the equations of motion, and the energymomentum tensor as part of taking the derivative of the Lagrangian analogousto the first step in Eq. (142). The gauge covariant derivative D µ is used sincethe F aµν field tensor is not gauge invariant. D µ L gauge = 2 D µ (cid:0) F aκη (cid:1) F κηa (C3)1Apply the useful relation for antisymmetric tensors where indicates thetensor dual (see Appendix: Definition of the Tensor Dual) of F κη , and G κη isan arbitrary antisymmetric second order tensor.( ∇ η F κµ ) G κη − ( ∇ µ F κη ) G κη = ˜ G µλ (cid:16) ∇ ν ˜ F νλ (cid:17) (C4)Move indices, and apply to our case where G = F ( ∇ µ F κη ) F κη = 2 ( ∇ η F κµ ) F κη − F µλ (cid:16) ∇ ν ˜ F νλ (cid:17) (C5)then substitute this relation into Eq. (C3), realizing that the same relationholds for second order tensors with a parameter index. D µ L = 4 ∇ η (cid:0) F aκµ (cid:1) F κηa − F aµλ ∇ ν (cid:16) ˜ F νλa (cid:17) (C6)In first term on the right, pull the second field tensor into the derivative, thensubtract the additional term arising from the product rule. D µ L = 4 D η (cid:0) F aκµ F κηa (cid:1) − F aκµ D η ( F κηa ) − F aµλ D ν (cid:16) ˜ F νλa (cid:17) (C7)Use the Kronecker delta tensor, and the definition of the Lagrangian. D η (cid:2) δ ηµ L − F aκµ F κηa (cid:3) = − F aκµ D η ( F κηa ) − ˜ F aµλ D ν (cid:16) ˜ F νλa (cid:17) (C8)Substitute the Lagrangian for the following definition of the energy momentumtensor, T ηµ (gauge) = δ ηµ F κλa F aκλ − F aκµ F κηa (C9)and Maxwell’s (symmetrized) equations, j κa = D η ( F ηκa )˜ j λa = D ν (cid:16) ˜ F νλa (cid:17) (C10)to get the divergence of the EMT for the gauge tensor field. ∇ ν T νµ (gauge) = F aνµ j νa + ˜ F aνµ ˜ j νa (C11)In this symmetrized form, ˜ j λ is regarded as a “magnetic” current of which is2set to zero to obtain Maxwell’s equations for electromagnetics.Notice that we can add the two “divergence of the EMT” Eqs. (C11) and147. Assume the Euler-Lagrange equation is satisfied for the wave function,Eq. (136) ∇ ν (cid:16) T νµ (gauge) + T µν (cid:17) = F aνµ j νa + ˜ F aνµ ˜ j νa + F aµν J νa (C12)Define the sum of the EMTs. T νµ (total) = T νµ (gauge) + T µν (C13)It is tempting to make the following identifications,˜ j νa = 0 j νa = − J νa (C14)so that the sum of the EMTs has a vanishing divergence. ∇ ν T νµ (total) = 0 (C15)We have constructed an isolated system with these identifications. The gaugefield is no longer external with the introduction of a Lagrangian Eq. (C2) forthat field. In this model, the wave function creates a charged current whichis subject to Lorentz forces applied by the gauge field, and at the same timethe charged current is the source of the gauge field via Eq. (C10). Appendix D: Definition of the Tensor Dual
Here we pause and define the “tensor dual” which is required for theLorentz force relation. The definition of the “ordinary” tensor dual is,˜ F µν =
12 1 √− g ε µνρσ F ρσ ˜ F ρσ = − √− gε ρσαβ F αβ (D1)where antisymmetric permutation tensors are defined , ε = 1 (D2)3Application of the dual twice, results in the negative of the original antisym-metric tensor for the space-time metric which has a negative determinant. − F µν =˜˜ F µν (D3) ∗ [email protected] Ian, J. R. Aitchison,
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Classical Fields: General Relativity and Gauge Theory (Wiley, New York,1982), p. 56, Eq. (2.7.28). link Albert Messiah,
Quantum Mechanics . Vol II (John Wiley & Sons, New York, 1966), p. 886,Eq. XX.30. The Klein-Gordon equation. Michael E. Peskin, Daniel V. Schroeder,
An Introduction to Quantum Field Theory (West-view Press, Boulder Colorado, 1995), Ch. 4.1 Perturbation Theory, Eq. (4.6) Paul H. Frampton,
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Classical Fields: General Relativity and Gauge Theory (Wiley, New York,1982), p. 109, Eq. (3.4.18) Asim O. Barut,
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Field Theory: A Modern Primer , (Addison-Wesley, Reading, MA, 1989),page 192, Eq. (6.2.4). Julian Schwinger,
Particles, Sources and Fields (Addison-Wesley Publishers, Inc. NewYork, 1970), p. 1, Section 1-1 “Unitary Transformations”, Eq. 1-1.2 Pierre Ramond,
Field Theory: A Modern Primer , (Addison-Wesley, Reading, MA, 1989),p. 189, Eq. (6.1.38). Pierre Ramond, Field Theory: A Modern Primer, (AddisonWesley, Reading, MA 1989), p.17 Michael E. Peskin, Daniel V. Schroeder, An Introduction to Quantum Field Theory (West-view Press, Perseus Books Group, 1995), Section 2.3 “The Klein-Gordon Field as HarmonicOscillators” Pierre Ramond,
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The Quantum Theory of Fields , Vol II (Cambridge University Press,Cambridge, 1996), p. 4 Eq. (15.1.10). Pierre Ramond,
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Torsion gravity (Reports on Progress in Physics, 2002, vol 65), p.601, Eqs. (1), (3), and (4). Online at . Pierre Ramond,
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Symmetries . Steven Weinberg,
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Particles, Sources and Fields (Addison-Wesley Publishers, Inc. NewYork, 1970), p. 3, view the paragraph preceding Eq. (1-1.26) Steven Weinberg,
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Lie Groups, Lie Algebras, and Some of Their Applications (John Wiley& Sons, New York, 1974), p. 100, Eq. (3.1) Robert Gilmore,
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The Quantum Theory of Fields , Vol II (Cambridge University Press,Cambridge, 1996), p. 6. “There is a deep analogy between the construction here of ob-jects that transform simply under gauge transformations and the construction in generalrelativity of objects that transform covariantly under general coordinate transformations.” Moshe Carmeli,
Classical Fields: General Relativity and Gauge Theory , (Wiley, New York,1982), p. 53, Eq. (2.7.10) Pierre Ramond,
Field Theory: A Modern Primer , (Addison-Wesley, Reading, MA 1989),p. 195, Eq. (6.2.13). In our case, B µ = 0. Moshe Carmeli,
Classical Fields: General Relativity and Gauge Theory , (Wiley, New York,1982), p. 9, Eq. (1.67). Pierre Ramond,
Field Theory: A Modern Primer , (Addison-Wesley, Reading, MA 1989),p. 194, Eq. (6.2.27) Robert Gilmore, Lie Groups, Physics, and Geometry, Cambridge University Press, 2008,p. 65, Eq (4.37). Steven Weinberg,
The Quantum Theory of Fields , vol II (Cambridge University Press,1996), p. 9, Eq. (15.2.4) Pierre Ramond,
Field Theory: A Modern Primer , (Addison-Wesley, Reading, MA 1989),p. 195, Eq. (6.2.34) Milutin Blagojevic,
Gravitation and Gauge Symmetries (Institute of Physics Publishing,Philadelphia, 2002), p. 377, following Eq. (A-16) Clinton L. Lewis,
Explicit Gauge Covariant Euler-Lagrange Equation , (Am. J. Phys. 77(9), September 2009). David Lovelock, Hanno Rund,
Tensors, Differential Forms, and Variational Principles (John Wiley & Sons, New York), p. 185, Eq. (1.18). The convention of covariant andcontravariant are swapped. L D Landau and E M Lifshitz,
The Classical Theory of Fields , 4th edition (PergamonPress, Reprinted (with corrections) 1987), p. 77, Eq. 32.3. Michio Kaku,
Quantum Field Theory, A Modern Introduction (Oxford University Press,New York, 1993), p. 27, Eq. (1.65) Robert M. Wald, General Relativity (University of Chicago Press, Chicago, 1984), p. 457,Eq. (E.1.36). L D Landau and E M Lifshitz,
The Classical Theory of Fields , 4th edition (PergamonPress, Reprinted (with corrections) 1987), p. 83, Eq. 33.9 Moshe Carmeli,
Classical Fields: General Relativity and Gauge Theory , (John Wiley &Sons, New York, 1982), p. 60, Problem (2.7.5) Julian Schwinger,
Particles, Sources and Fields (Addison-Wesley Publishers, Inc. NewYork, 1970), p. 237 Eq. (3-8.90). Moshe Carmeli,
Classical Fields: General Relativity and Gauge Theory , (Wiley, New York,1982), p. 38, Eqs. (2.5.19) and (2.5.20). Tensor duals remain a tensor of weight 0. David Lovelock, Hanno Rund,