Emerging of massive gauge particles in inhomogeneous local gauge transformations

Transcription

1 Emerging of massive gauge particles in inhomogeneous local gauge transformations Struckmeier GSI Helmholtzentrum für Schwerionenforschung GmbH, Planckstr. 1, D Darmstadt Goethe University, Max-von-aue-Str. 1, D Frankfurt am Main, Germany Abstract. A generalised theory of gauge transformations is presented on the basis of the covariant Hamiltonian formalism of field theory, for which the covariant canonical field equations are equivalent to the Euler-agrange field equations. Similar to the canonical transformation theory of point dynamics, the canonical transformation rules for fields are derived from generating functions. Thus in contrast to the usual agrangian description the covariant canonical transformation formalism automatically ensures the mappings to preserve the action principle, and hence to be physical. On that basis, we work out the theory of inhomogeneous local gauge transformations that generalises the conventional local SUN gauge transformation theory. It is shown that massive gauge bosons naturally emerge in this description, which thus could supersede the Higgs mechanism. Die Fruchtbarkeit des neuen Gesichtspunktes der Eichinvarianz hätte sich vor allem am Problem der Materie zu zeigen. Weyl 1919 [1] The fruitfulness of the new viewpoint of gauge invariance would have to show up in particular on the problem of matter. 1. Introduction The principle of local gauge invariance has been proven to be an eminently fruitful device for deducing all elementary particle interactions within the standard model. On the other hand, the gauge principle is justified only as far as it works : a deeper rationale underlying the gauge principle apparently does not exist. In this respect, the gauge principle corresponds to other basic principles of physics, such as Fermat s principle of least time, the principle of least action as well as its quantum generalisation leading to Feynman s path integral formalism. The failure of the conventional gauge principle to explain the existence of massive gauge bosons has led to supplementing it with the Higgs-Kibble mechanism [2, 3]. An alternative strategy to resolve the mass problem would be to directly generalise the conventional gauge principle in a natural way. One way to achieve this was to require the system s covariant Hamiltonian to be form-invariant not only under unitary transformations of the fields in iso-space, but also under variations of the space-time metric. This idea of a generalisation of the conventional gauge principle has been successfully worked out and was

2 2 Inhomogeneous local gauge transformations published recently [4]. In this description, the gauge field causes a non-vanishing curvature tensor, and this curvature tensor appears in the field equations as a mass factor. With the actual paper, a second natural generalisation of the conventional gauge transformation formalism will be presented that extends the conventional SUN gauge theory to include inhomogeneous linear mappings of the fields. As it turns out, the local gaugeinvariance of the system s agrangian then requires the existence of massive gauge fields, with the mass playing the role of a second coupling constant. We thereby tackle the longstanding inconsistency of the conventional gauge principle that requires gauge bosons to be massless in order for any theory to be locally gauge-invariant. This will be achieved without postulating a particular potential function Mexican hat and without requiring a symmetry breaking phenomenon. Conventional gauge theories are commonly derived on the basis of agrangians of relativistic field theory cf, for instance, [5, 6, 7]. Although perfectly valid, the agrangian formulation of gauge transformation theory is not the optimum choice. The reason is that in order for a agrangian transformation theory to be physical, hence to maintain the action principle, it must be supplemented by additional structure, referred to as the minimum coupling rule. In contrast, the formulation of gauge theories in terms of covariant Hamiltonians each of them being equivalent to a corresponding agrangian may exploit the framework of the canonical transformation formalism. With the transformation rules for all fields and their canonical conjugates being derived from generating functions, we restrict ourselves from the outset to exactly the subset of transformations that preserve the action principle, hence ensure the actual gauge transformation to be physical. No additional structure needs to be incorporated for setting up an amended Hamiltonian that is locally gauge-invariant on the basis of a given globally gauge-invariant Hamiltonian. The covariant derivative defined by the minimum coupling rule automatically arises as the respective canonical momentum. Furthermore, it is no longer required to postulate the field tensor to be skew-symmetric in its space-time indices as this feature directly emerges from the canonical transformation formalism. Prior to working out the inhomogeneous local gauge transformation theory in the covariant Hamiltonian formalism the latter dating back to DeDonder [8] and Weyl [9] a concise review of the concept of covariant Hamiltonians in local coordinate representation is outlined in section 2. Thereafter, the canonical transformation formalism in the realm of field theory is sketched briefly in section 3. In these sections, we restrict our presentation to exactly those topics of the canonical formalism that are essential for working out the inhomogeneous gauge transformation theory, which will finally be covered in section 4. The requirement of inhomogeneous local gauge invariance naturally generalises the conventional SUN gauge principle cf, for instance, [10], where the form-invariance of the covariant Hamiltonian density is demanded under homogeneous unitary mappings of the fields in iso-space. In the first step, a generating function of type F 2 is set up that merely describes the demanded transformation of the fields in iso-space. As usual, this transformation forces us to introduce gauge fields that render an appropriately amended Hamiltonian locally

3 Inhomogeneous local gauge transformations 3 gauge-invariant if the gauge fields follow a particular transformation law. In our case of an inhomogeneous mapping, we are forced to introduce two independent sets of gauge fields, each of them requiring its own transformation law. In the second step, an amended generating function F 2 is constructed in a way to define these transformation laws for the two sets of gauge fields in addition to the rules for the base fields. As the characteristic feature of the canonical transformation formalism, this amended generating function also provides the transformation law for the conjugates of the gauge fields and for the Hamiltonian. This way, we derive the Hamiltonian that is form-invariant under both the inhomogeneous mappings of the base fields as well as under the required mappings of the two sets of gauge fields. In a third step, it must be ensured that the canonical field equations emerging from the gauge-invariant Hamiltonian are consistent with the canonical transformation rules. As usual in gauge theories, the Hamiltonian must be further amended by terms that describe the freefield dynamics of the gauge fields while maintaining the overall form-invariance of the final Hamiltonian. Amazingly, this also works for our inhomogeneous gauge transformation theory and uniquely determines the final gauge-invariant Hamiltonian H 3. The Hamiltonian H 3 is then egendre-transformed to yield the equivalent gaugeinvariant agrangian density 3. The latter can then serve as the starting point to set up the Feynman diagrams for the various mutual interactions of base and gauge fields. As examples, the locally gauge-invariant agrangians that emerge from base systems of N-tuples of massless spin-0 and massive spin- 1 fields are presented Covariant Hamiltonian density In field theory, the Hamiltonian density is usually defined by performing an incomplete egendre transformation of a agrangian density that only maps the time derivative t φ of a field φt, x, y, z into a corresponding canonical momentum variable, π t. Taking then the spatial integrals results in a description that corresponds to that of non-relativistic Hamiltonian point dynamics. Yet, in analogy to relativistic point dynamics [11], a covariant Hamiltonian description of field theory must treat space and time variables on equal footing. If is a orentz scalar, this property is passed to the covariant DeDonder-Weyl Hamiltonian density H that emerges from a complete egendre transformation of. Moreover, this description enables us to devise a consistent theory of canonical transformations in the realm of classical field theory Covariant canonical field equations The transition from particle dynamics to the dynamics of a continuous system is based on the assumption that a continuum limit exists for the given physical problem cf, for instance, [12]. This limit is defined by letting the number of particles involved in the system increase over all bounds while letting their masses and distances go to zero. In this limit, the information on the location of individual particles is replaced by the value of a smooth

4 4 Inhomogeneous local gauge transformations function φx that is given at a spatial location x 1, x 2, x 3 at time t x 0. In this notation, the index µ runs from 0 to 3, hence distinguishes the four independent variables of spacetime x µ x 0, x 1, x 2, x 3 t, x, y, z, and x µ x 0, x 1, x 2, x 3 t, x, y, z. We furthermore assume that the given physical problem can be described in terms of a set of I = 1,..., N possibly interacting scalar fields φ I x, with the index I enumerating the individual fields. A transformation of the fields in iso-space is not associated with any nontrivial metric. We, therefore, do not use superscripts for these indices as there is not distinction between covariant and contravariant components. In contrast, Greek indices are used for those components that are associated with a metric such as the derivatives with respect to a spacetime variable, x µ. Throughout the article, the summation convention is used. Whenever no confusion can arise, we omit the indices in the argument list of functions in order to avoid the number of indices to proliferate. The agrangian description of the dynamics of a continuous system is based on the agrangian density function that is supposed to carry the complete information on the given physical system. In a first-order field theory, the agrangian density is defined to depend on the φ I, possibly on the vector of independent variables x µ, and on the four first derivatives of the fields φ I with respect to the independent variables, i.e., on the 1-forms covectors µ φ I t φ I, x φ I, y φ I, z φ I. The Euler-agrange field equations are then obtained as the zero of the variation δs of the action integral S = φ I, µ φ I, x d 4 x 1 as φ I = 0. 2 φ I To derive the equivalent covariant Hamiltonian description of continuum dynamics, we first define for each field φ I x a 4-vector of conjugate momentum fields π µ I x. Its components are given by π µ I = µ φ I φ I. 3 The 4-vector π µ I is thus induced by the agrangian as the dual counterpart of the 1-form µ φ I. For the entire set of N scalar fields φ I x, this establishes a set of N conjugate 4-vector fields. With this definition of the 4-vectors of canonical momenta π I x, we can now define the Hamiltonian density Hφ I, π I, x as the covariant egendre transform of the agrangian density φ I, µ φ I, x Hφ I, π I, x = π φ x φ I, µ φ I, x. 4 In order for the Hamiltonian H to be valid, we must require the egendre transformation to be regular, which means that for each index I the Hesse matrices 2 / µ φ I ν φ I are non-singular. This ensures that by means of the egendre transformation, the Hamiltonian H

5 Inhomogeneous local gauge transformations 5 takes over the complete information on the given dynamical system from the agrangian. The definition of H by Eq. 4 is referred to in literature as the DeDonder-Weyl Hamiltonian density. Obviously, the dependencies of H and on the φ I and the x µ only differ by a sign, H = expl, expl H = = φ I φ I φ I = π I x. These variables thus do not take part in the egendre transformation of Eqs. 3, 4. Thus, with respect to this transformation, the agrangian density represents a function of the µ φ I only and does not depend on the canonical momenta π µ I, whereas the Hamiltonian density H is to be considered as a function of the π µ I only and does not depend on the derivatives µφ I of the fields. In order to derive the second canonical field equation, we calculate from Eq. 4 the partial derivative of H with respect to π µ I, H π µ I = δ I δ µ φ = φ I µ φ I = π δ I δ µ = π µ I. The complete set of covariant canonical field equations is thus given by H π µ I = φ I, H φ I = π I. 5 This pair of first-order partial differential equations is equivalent to the set of second-order differential equations of Eq. 2. We observe that in this formulation of the canonical field equations, all coordinates of space-time appear symmetrically similar to the agrangian formulation of Eq. 2. Provided that the agrangian density is a orentz scalar, the dynamics of the fields is invariant with respect to orentz transformations. The covariant egendre transformation 4 passes this property to the Hamiltonian density H. It thus ensures a priori the relativistic invariance of the fields that emerge as integrals of the canonical field equations if and hence H represents a orentz scalar. 3. Canonical transformations in covariant Hamiltonian field theory The covariant egendre transformation 4 allows us to derive a canonical transformation theory in a way similar to that of point dynamics. The main difference is that now the generating function of the canonical transformation is represented by a vector rather than by a scalar function. The main benefit of this formalism is that we are not dealing with arbitrary transformations. Instead, we restrict ourselves right from the beginning to those transformations that preserve the form of the action functional. This ensures all eligible transformations to be physical. Furthermore, with a generating function, we not only define the transformations of the fields but also pinpoint simultaneously the corresponding transformation law of the canonical momentum fields.

6 6 Inhomogeneous local gauge transformations 3.1. Generating functions of type F 1 φ, Φ, x Similar to the canonical formalism of point mechanics, we call a transformation of the fields φ, π Φ, Π canonical if the form of the variational principle that is based on the action functional 1 is maintained, δ πi φ I Hφ, π, x d 4 x =! δ Π Φ I R x I R x H Φ, Π, x d 4 x. 6 Equation 6 tells us that the integrands may differ by the divergence of a vector field F µ 1, whose variation vanishes on the boundary R of the integration region R within space-time δ R F 1 d4 x = δ R F 1 ds! = 0. The immediate consequence of the form invariance of the variational principle is the form invariance of the covariant canonical field equations 5 H Π µ I = Φ I, H Φ I = Π I. For the integrands of Eq. 6 hence for the agrangian densities and we thus obtain the condition = + F 1 πi φ I x Hφ, π, x = Φ I Π I x H Φ, Π, x + F 1 x. 7 With the definition F µ 1 F µ 1 φ, Φ, x, we restrict ourselves to a function of exactly those arguments that now enter into transformation rules for the transition from the original to the new fields. The divergence of F µ 1 writes, explicitly, F1 x = F 1 φ I φ I x + F 1 Φ I Φ I x + F 1. 8 expl The rightmost term denotes the sum over the explicit dependence of the generating function F µ 1 on the x ν. Comparing the coefficients of Eqs. 7 and 8, we find the local coordinate representation of the field transformation rules that are induced by the generating function F µ 1 π µ I = F µ µ 1, Π µ I φ = F 1, H = H + F 1 I Φ I. 9 expl The transformation rule for the Hamiltonian density implies that summation over is to be performed. In contrast to the transformation rule for the agrangian density of Eq. 7, the rule for the Hamiltonian density is determined by the explicit dependence of the generating function F µ 1 on the x ν. Hence, if a generating function does not explicitly depend on the independent variables, x ν, then the value of the Hamiltonian density is not changed under the particular canonical transformation emerging thereof. Differentiating the transformation rule for π µ I with respect to Φ, and the rule for Π µ with respect to φ I, we obtain a symmetry relation between original and transformed fields π µ I Φ = 2 F µ 1 φ I Φ = Πµ φ I.

7 Inhomogeneous local gauge transformations 7 The emerging of symmetry relations is a characteristic feature of canonical transformations. As the symmetry relation directly follows from the second derivatives of the generating function, is does not apply for arbitrary transformations of the fields that do not follow from generating functions Generating functions of type F 2 φ, Π, x The generating function of a canonical transformation can alternatively be expressed in terms of a function of the original fields φ I and of the new conjugate fields Π µ I. To derive the pertaining transformation rules, we perform the covariant egendre transformation µ F µ 2 φ, Π, x = F µ 1 φ, Φ, x + Φ Π µ, Πµ I = F Φ I By definition, the functions F µ 1 and F µ 2 agree with respect to their φ I and x µ dependencies F µ 2 φ I = F µ 1 φ I = π µ I, F 2 = F 1 expl = H H. expl The variables φ I and x µ thus do not take part in the egendre transformation from Eq. 10. Therefore, the two F µ 2 -related transformation rules coincide with the respective rules derived previously from F µ 1. As F µ 1 does not depend on the Π µ I whereas F µ 2 does not depend on the the Φ I, the new transformation rule thus follows from the derivative of F µ 2 with respect to Π ν as F µ 2 Π µ = Φ Π ν = Φ I Π ν δ I δ ν µ. I We thus end up with set of transformation rules π µ I = F µ 2, Φ I δ ν µ = F µ 2, H = H + F 2 φ I Π ν I, 11 expl which is equivalent to the set 9 by virtue of the egendre transformation 10 if the matrices 2 F µ 1 / φ I Φ are non-singular. From the second partial derivations of F µ 2 one immediately derives the symmetry relation π µ I Π ν = 2 F µ 2 φ I Π ν = Φ φ I δ µ ν, whose existence characterises the transformation to be canonical Gauge theories as canonical transformations Devising gauge theories in terms of canonical transformations turns out to be a particularly useful application of the canonical formalism in the realm of classical field theory. The systematic procedure to pursue is as follows: i Construct the generating function F µ 2 that defines the desired local transformation of the fields of the given covariant system Hamiltonian H. If the given system is described in terms of a agrangian, the corresponding Hamiltonian H is obtained by a covariant egendre transformation according to Eq. 4.

8 8 Inhomogeneous local gauge transformations ii Calculate the divergence of F µ 2 to find the transformation rule for the Hamiltonian H. iii Introduce the appropriate gauge field Hamiltonian H g that is eligible to compensate the terms of the divergence of F µ 2. iv Derive the transformation rules for the gauge fields from the requirement that the amended Hamiltonian H 1 = H + H g be form-invariant. v Construct the amended generating function F µ 2 fields and gauge fields. that defines the transformation of base vi Calculate the divergence of F µ 2 to find the transformation rule for the amended Hamiltonian H 1. vii Express the divergence of F µ 2 in terms of the physical fields and their conjugates making use of their transformation rules. viii Provided that all terms come up in pairs, i.e., if they have the same form in the original and in the transformed field variables, this uniquely determines the form of the Hamiltonian H 2 that is locally form-invariant. ix Add the Hamiltonian H kin describing the kinetics of the free gauge fields. It must be ensured that H kin is also form-invariant under the given transformation rules to maintain the local form-invariance of the final Hamiltonian H 3 = H 2 + H kin. x Optionally egendre-transform the final Hamiltonian H 3 to determine the corresponding locally gauge-invariant agrangian 3. We will follow this procedure in the next section to work out a agrangian 3 that is forminvariant under an inhomogeneous local gauge transformation. 4. General inhomogeneous local gauge transformation As a generalisation of the homogeneous local UN gauge group, we now treat the corresponding inhomogeneous gauge group for the case of an N-tuple of fields φ I External gauge fields We consider a system consisting of an N-tuple φ of complex fields φ I with I = 1,..., N, and φ its adjoint, φ = φ 1. φ N, φ = φ 1 φ N. A general inhomogeneous linear transformation may be expressed in terms of a complex matrix Ux = u I x, U x = u I x and a vector ϕx = ϕ I x that generally depend explicitly on the independent variables, x µ, as Φ = U φ + ϕ, Φ = φ U + ϕ Φ I = u I φ + ϕ I, Φ I = φ u I + ϕ I. 12

9 Inhomogeneous local gauge transformations 9 With this notation, φ I stands for a set of I = 1,..., N complex fields φ I. In other words, U is supposed to define an isomorphism within the space of the φ I, hence to linearly map the φ I into objects of the same type. The quantities ϕ I x have the dimension of the base fields φ I and define a local shifting transformation of the Φ I in iso-space. Physically, this means that the system is now required to be form-invariant both under local unitary transformations in iso-space and under local variations of background fields ϕ I x. The transformation 12 follows from a generating function that corresponding to H must be a real-valued function of the generally complex fields φ I and their canonical conjugates, π µ I, F µ 2 φ, φ, Π µ, Π µ, x = Π µ U φ + ϕ + φ U + ϕ Π µ = Π µ K uk φ + ϕ K + φk u K + ϕ Π µ. 13 According to Eqs. 11 the set of transformation rules follows as π µ I = F µ 2 φ I = Π µ Ku K δ I, Φ I δ µ ν = F µ 2 Π ν I = φ K u K + ϕ δ µ ν δ I π µ I = F µ 2 = δ IK u K Π µ φ, Φ Iδ ν µ = F µ 2 I Π ν = δ ν µ δ IK uk φ + ϕ K. I The complete set of transformation rules and their inverses then read in component notation Φ I = u I φ + ϕ I, Φ I = φ u I + ϕ I, Π µ I = u I π µ, Πµ I = π µ u I φ I = u I Φ ϕ, φi = Φ ϕ ui, π µ I = u I Π µ, πµ I = Πµ u I. 14 We restrict ourselves to transformations that preserve the contraction π π Π Π = π U U π = π π = U U = 1 = UU Π I Π I = π u I u IK π K = π Kπ K = u I u IK = δ K = u I u IK. This means that U = U 1, hence that the matrix U is supposed to be unitary. As a unitary matrix, Ux is a member of the unitary group UN U x = U 1 x, det Ux = 1. For det Ux = +1, the matrix Ux is a member of the special group SUN. We require the Hamiltonian density H to be form-invariant under the global gauge transformation 12, which is given for U, ϕ = const., hence for all u I, ϕ I not depending on the independent variables, x µ. Generally, if U = Ux, ϕ = ϕx, then the transformation 14 is referred to as a local gauge transformation. The transformation rule for the Hamiltonian is then determined by the explicitly x µ -dependent terms of the generating function F µ 2 according to H H = F 2 = Π I expl ui x φ + ϕ I u I + φ I = π ui K u KI x φ + ϕ I = π K φ φ K π uki u I + φ I u I + ϕ Π + ϕ u K π K + π I u I ϕ + ϕ u Iπ I. 15

10 10 Inhomogeneous local gauge transformations In the last step, the identity u I x u u IK µ IK + u I x = 0 µ was inserted. If we want to set up a Hamiltonian H 1 that is form-invariant under the local, hence x µ -dependent transformation generated by 13, then we must compensate the additional terms 15 that emerge from the explicit x µ -dependence of the generating function 13. The only way to achieve this is to adjoin the Hamiltonian H of our system with terms that correspond to 15 with regard to their dependence on the canonical variables, φ, φ, π µ, π µ. With a unitary matrix U, the u I -dependent terms in Eq. 15 are skew- Hermitian, u KI u I = u I u IK = u I u IK, or in matrix notation U U = U U U = U x µ x, µ u KI u I = u I u IK = u I u IK, U x U = U U µ x = U µ x U. µ The u KI u I / -dependent terms in Eq. 15 can thus be compensated by a Hermitian matrix a K of 4-vector gauge fields, with each off-diagonal matrix element, a K, K, a complex 4-vector field with components a Kµ, µ = 0,..., 3 u KI u I a Kµ, a Kµ = a Kµ = a Kµ. Correspondingly, the term proportional to u I ϕ / is compensated by the µ-components M I b µ of a vector M I b of 4-vector gauge fields, u I ϕ M I b µ, ϕ u I b µ M I. The term proportional to ϕ / x u I is then compensated by the adjoint vector b M I. The dimension of the constant real matrix M is [M] = 1 and thus has the natural dimension of mass. The given system Hamiltonian H must be amended by a Hamiltonian H a of the form H 1 = H + H a, H a = ig π Kφ φ K π ak + π I M I b + b M I π I 16 in order for H 1 to be form-invariant under the canonical transformation that is defined by the explicitly x µ -dependent generating function from Eq. 13. With a real coupling constant g, the gauge Hamiltonian H a is thus real. Submitting the amended Hamiltonian H 1 to the canonical transformation generated by Eq. 13, the new Hamiltonian H 1 emerges as H 1 = H 1 + F 2 = H + H a + F 2 expl expl u I ig a K + u KI = H + π Kφ φ K π + π I M I b + u I ϕ + x b M I + ϕ x u I π I! = H + ig Π KΦ Φ K Π AK + Π I M I B + B M I Π I,

11 Inhomogeneous local gauge transformations 11 with the A Iµ and B Iµ defining the gauge field components of the transformed system. The form of the system Hamiltonian H 1 is thus maintained under the canonical transformation, H 1 = H + H a, H a = ig Π KΦ Φ K Π AK + Π I M I B + B M I Π I, 17 provided that the given system Hamiltonian H is form-invariant under the corresponding global gauge transformation 14. In other words, we suppose the given system Hamiltonian Hφ, φ, π µ, π µ, x to remain form-invariant if it is expressed in terms of the transformed fields, H Φ, Φ, Π µ, Π µ global GT, x = Hφ, φ, π µ, π µ, x. Replacing the transformed base fields by the original ones according to Eqs. 14, the gauge fields must satisfy the condition π K φ φ K π u I ig a K + u KI x + π ϕ I M I b + u I + b M I + ϕ x u I πi = ig π I u IK u φ + π I u IK ϕ φ u K u I π I ϕ K u I π I AK + π I u IK M K B + B M K u KI π I, which yields with Eqs. 14 the following inhomogeneous transformation rules for the gauge fields a K, b, and b by comparing the coefficients that are associated with the independent dynamical variables π µ I, π µ I, π µ I φ, and φ π µ I A Kµ = u K a Iµ u I + 1 ig B µ = M I B µ = u KI u I u IK M K b µ ig A IKµ ϕ K + ϕ I x µ b µ M K u KI + ig ϕ K A KIµ + ϕ I M I. Herein, M denotes the inverse matrix of M, hence MK M I = M K MI = δ KI. We observe that for any type of canonical field variables φ I and for any Hamiltonian system H, the transformation of both the matrix a I as well as the vector b I of 4-vector gauge fields is uniquely determined according to Eq. 18 by the unitary matrix Ux and the translation vector ϕx that determine the local transformation of the N base fields φ. In a more concise matrix notation, Eqs. 18 are A µ = U a µ U + 1 ig MB µ U U 18 = UM b µ ig A µ ϕ + ϕ 19 B µ M T = b µ M T U + ig ϕ A µ + ϕ x. µ Inserting the transformation rules for the base fields, Φ = Uφ + ϕ and Φ = φ U + ϕ into Eqs. 19, we immediately find the homogeneous transformation conditions Φ ig A µ Φ MB µ = U φ ig a µ φ Mb µ

12 12 Inhomogeneous local gauge transformations Φ φ x + ig ΦA µ µ B µ M T = x + ig φa µ µ b µ M T U. We identify the amended partial derivatives as the covariant derivative that defines the minimum coupling rule for our inhomogeneous gauge transformation. It reduces to the conventional minimum coupling rule for the homogeneous gauge transformation, hence for ϕ 0, M Including the gauge field dynamics With the knowledge of the required transformation rules for the gauge fields from Eq. 18, it is now possible to redefine the generating function 13 to also describe the gauge field transformations. This simultaneously defines the transformations of the canonical conjugates, p µν K and qµν, of the gauge fields a Kµ and b µ, respectively. Furthermore, the redefined generating function yields additional terms in the transformation rule for the Hamiltonian. Of course, in order for the Hamiltonian to be invariant under local gauge transformations, the additional terms must be invariant as well. The transformation rules for the base fields φ I and the gauge fields a I, b I Eq. 18 can be regarded as a canonical transformation that emerges from an explicitly x µ -dependent and real-valued generating function vector of type F µ 2 F µ 2 = F µ 2 φ, φ, Π, Π, a, P, b, b, Q, Q, x, = Π µ K uk φ + ϕ K + φk u K + ϕ Π µ 20 + P µ K + ig M Q µ ϕ K ig ϕ Q µ M K u KN a NI u I + 1 u KI ig x u I + Q µ M K u KI M I b + ϕ K + b K M IK u I + ϕ M Q µ. With the first line of 20 matching Eq. 13, the transformation rules for canonical variables φ, φ and their conjugates, π µ, π µ, agree with those from Eqs. 14. The rules for the gauge fields A K, B K, and B K emerge as A K δ µ ν = F µ 2 P ν K B δ µ ν = F µ 2 Q ν B δ µ ν = F µ 2 Q ν u KN a NI u I + 1 u KI ig = δ µ ν M K [ u KI M I b + ϕ K = δ µ ν M K u KI M I b + ϕ K = δ ν µ x u I ig u KN a NI u I + u ] KI x u I ϕ x ig A Kϕ [ = δ ν µ b K M IK u I + ϕ x + ϕ K ig u KN a NI u I + u ] KI x u I M = δ ν µ b K M IK u I + ϕ x + ig ϕ K A K M, which obviously coincide with Eqs. 18 as the generating function 20 was devised accordingly. The transformation of the conjugate momentum fields is obtained from the

13 Inhomogeneous local gauge transformations 13 generating function 20 as q νµ = F µ 2 = M I u IK MK Q νµ b, MK Q νµ K = u I ν M KI q νµ K Thus, the expression q νµ = F µ 2 b ν = Q νµ M K u KI M I, Q νµ K M K = q νµ K M KI u I 21 p νµ IN = F µ 2 = u I P νµ K a + ig M Q νµ ϕ K ig ϕ Q νµ M K u KN NIν = u I P νµ K + ig M Q νµ Φ K ig Φ Q νµ M K u KN ig M I q νµ φ N + ig φ I q νµ M N. M N p νµ IN + ig M I q νµ φ N ig φ I q νµ = u I P νµ K + ig M Q νµ Φ K ig Φ Q νµ M K u KN 22 transforms homogeneously under the gauge transformation generated by Eq. 20. The same homogeneous transformation law holds for the expression f Iµν = a Iν x a Iµ + ig a µ x ν IKν a Kµ a IKµ a Kν = u IK F Kµν u 23 F Iµν = A Iν x A Iµ + ig A µ x ν IKν A Kµ A IKµ A Kν, which directly follows from the transformation rule 18 for the gauge fields a Iµ. Making use of the initially defined mapping of the base fields 12, the transformation rule 18 for the gauge fields b Kµ, b Kµ is converted into Φ φ ig A KµΦ K M K B Kµ = u Φ + ig Φ KA Kµ B Kµ M K = x ig a Kµφ µ K M K b Kµ φ x + ig φ Ka µ Kµ b Kµ M K u. 24 The above transformation rules can also be expressed more clearly in matrix notation q νµ = M T U M T Q νµ, M T Q νµ = U M T q νµ q νµ = Q νµ M U M, Q νµ M = q νµ M U p νµ = U P νµ + ig M T Q νµ ϕ ig ϕ Q νµ M f µν = U F µν U, and Φ ig A µ Φ MB µ Φ + ig ΦA µ B µ M T = U f µν = a ν a µ x ν + ig a ν a µ a µ a ν 25 φ = U x ig a µ µ φ M b µ φ x + ig φ a µ µ b µ M T U P νµ + ig M T Q νµ Φ ig Φ Q νµ M = U p νµ + ig M T q νµ φ ig φ q νµ M U.

14 14 Inhomogeneous local gauge transformations It remains to work out the difference of the Hamiltonians that are submitted to the canonical transformation generated by 20. Hence, according to the general rule from Eq. 11, we must calculate the divergence of the explicitly x µ -dependent terms of F µ 2 F 2 = Π K expl + P β + + Q β ukn uk φ + ϕ K + φ K u K + ϕ Π K + ig M Q β ϕ K ig ϕ Q β M K x a u I NIu β I + u KN a NI x + 1 u KI u I β ig x u KI β ig x u β I M Q β ϕ K x ϕ β Qβ M K ig u KN a NI u I + u KI x u I M uki K x M Ib β + 2 ϕ K u I + b K M IK x + 2 ϕ β M Q β. 26 We are now going to express all u I - and ϕ K -dependencies in 26 in terms of the field variables making use of the canonical transformation rules. To this end, the constituents of Eq. 26 are split into three blocks. The Π-dependent terms of can be converted this way by means of the transformation rules 14 and 18 Π uk K x uk = Π K φ + ϕ K + u K φ K x u IΦ I ϕ I + ϕ K + ϕ Π u K + ΦI ϕ I uik + ϕ Π = ig Π KΦ Φ K Π AK + Π KM K B + B K M K Π ig π Kφ φ K π ak π KM K b + b K M K π. 27 The second derivative terms in Eq. 26 are symmetric in the indices and β. If we split P β K and Q β into a symmetric P β K, Qβ and a skew-symmetric parts P [β] K, P [β] in and β P β K = P β K + P [β] K, Q β = Q β P [β] K = Q [β], Q [β] = 1 2 K P β K, P β P β Q β Qβ then the second derivative terms in Eq. 26 vanish for P [β] K P [β] K 2 u KI x = 0, β K ukn K = 1 2, Q β = 1 2 and Q[β], P β K + P β K Q β + Qβ 2 ϕ Q[β] = 0, Q [β] 2 ϕ K K x = 0. β By inserting the transformation rules for the gauge fields from Eqs. 18, the remaining terms of 26 for the skew-symmetric part of P β K are converted into P [β] + ig M Q [β] ϕ K ig ϕ Q [β] M K x β + a u I NIu I + u KN a NI x + 1 β ig M Q [β] ϕ K ϕ Q[β] u KI M K ig A K u I,

15 Inhomogeneous local gauge transformations 15 + Q [β] u KI M K x M u IK Ib β + b M I = 1 β ig P 2 K A KIA Iβ A KIβ A I + 1ig 2 1 ig Qβ 2 MK Q [β] B β M K A KI MI B M K A KIβ MI MI A IK M K B β M I A IKβ M K B + 1 ig pβ 2 K a KIa Iβ a KIβ a I 1ig b 2 β M K a KI MI b M K a KIβ MI + 1 ig qβ 2 For the symmetric parts of P β K P β + ig M Q β = K ukn x β + + Q β P β K + Q β + q β Q β MI a IK M K b β M I a IKβ M K b. 28 and Qβ, we obtain M K ϕ K ig ϕ Q β a u I NIu I + u K a I x + 1 β ig M K M Q β ϕ K ϕ Qβ M K uki M Ib + b M I u IK = 1P β AK 2 K 1 2 pβ K 2 ϕ K u KI ig A K + u I x u KI β ig x u β I b M I u IK + 2 ϕ K M K Q β + ig M Q β ϕ K ig ϕ Q β A K a I M K u K x u β I uki M K x M Ib β + 2 ϕ K x ig A ϕ β K x + 2 ϕ K β x + ig ϕ β x A β K M K Q β + A Kβ + 1 BK 2 Qβ K + B Kβ + 1 BK + B Kβ 2 ak + a Kβ 1 bk 2 qβ K x + b Kβ 1 bk β 2 x + b Kβ β Q β K q β K. 29 In summary, by inserting the transformation rules into Eq. 26, the divergence of the explicitly x µ -dependent terms of F µ 2 and hence the difference of transformed and original Hamiltonians can be expressed completely in terms of the canonical variables as F 2 = ig Π KΦ Φ K Π AK + Π KM K B + B K M K Π expl ig π Kφ φ K π ak π KM K b + b K M K π 1 β ig P 2 K A KIA Iβ A KIβ A I + 1 ig pβ 2 K a KIa Iβ a KIβ a I + 1ig B 2 β M K A KI MI B M K A KIβ MI Q β 1 ig Qβ 2 MI A IK M K B β M I A IKβ M K B

16 16 Inhomogeneous local gauge transformations 1ig b 2 β M K a KI MI b M K a KIβ MI q β MI a IK M K b β M I a IKβ M K b + 1 ig qβ 2 AK P β K 1 2 pβ K ak + A Kβ + a Kβ Qβ K 1 2 qβ K BK bk + B Kβ + b Kβ BK bk B Kβ + b Kβ Q β K q β K. We observe that all u I and ϕ I -dependencies of Eq. 26 were expressed symmetrically in terms of both the original and the transformed complex base fields φ, Φ and 4-vector gauge fields a K, A K,b, B, in conjunction with their respective canonical momenta. Consequently, an amended Hamiltonian H 2 of the form H 2 = Hπ, φ, x + ig π Kφ φ K π ak + π KM K b + b K M K π 1 ig pβ 2 K a KI a Iβ a KIβ a I pβ K + 1ig b 2 β M K a KI b M K a KIβ MI q β ig qβ M I aik M K b β a IKβ M K b qβ K bk + b Kβ bk is then transformed according to the general rule 11 H 2 = H 2 + F 2 into the new Hamiltonian expl ak + a Kβ + b Kβ q β K 30 H 2 = HΠ, Φ, x + ig Π KΦ Φ K Π AK + Π KM K B + B K M K Π 1 β ig P 2 K A KI A Iβ A KIβ A I + 1P β 2 K + 1ig B 2 β M K A KI B M K A KIβ MI Q β ig Qβ M I AIK M K B β A IKβ M K B Qβ K BK + B Kβ BK AK + A Kβ + B Kβ Q β K. 31 The entire transformation is thus form-conserving provided that the original Hamiltonian Hπ, φ, x is also form-invariant if expressed in terms of the new fields, HΠ, Φ, x = Hπ, φ, x, according to the transformation rules 14. In other words, Hπ, φ, x must be form-invariant under the corresponding global gauge transformation. As a common feature of all gauge transformation theories, we must ensure that the transformation rules for the gauge fields and their conjugates are consistent with the field equations for the gauge fields that follow from final form-invariant amended Hamiltonians, H 3 = H 2 + H kin and H 3 = H 2 + H kin. In other words, H kin and the form-alike H kin must be chosen in a way that the transformation properties of the canonical equations for

17 Inhomogeneous local gauge transformations 17 the gauge fields emerging from H 3 and H 3 are compatible with the canonical transformation rules 18. These requirements uniquely determine the form of both H kin and H kin. Thus, the Hamiltonians 30 and 31 must be further amended by kinetic terms H kin and H kin that describe the dynamics of the free 4-vector gauge fields, a K, b and A K, B, respectively. Of course, H kin must be form-invariant as well if expressed in the transformed dynamical variables in order to ensure the overall form-invariance of the final Hamiltonian. An expression that fulfils this requirement is obtained from Eqs. 21 and 22 H kin = 1 2 qβ q β 1 p β 4 I + ig M I q β φ ig φ I q β M p Iβ + ig M K q Kβ φ I ig φ q Kβ MKI. 32 The condition for the first term to be form-invariant is q β q β = Q β M K u KI M I M }{{ N} = det M 2 Q β = Q β Q β The mass matrix M must thus be orthogonal u NR! =δ IN det M 2 M K MK Q }{{} β! =δ det M 2 M SR Q Sβ M M T = 1 det M From H 3 and, correspondingly, from H 3, we will work out the condition for the canonical field equations to be consistent with the canonical transformation rules 18 for the gauge fields and their conjugates 21. With H kin from Eq. 32, the total amended Hamiltonian H 3 is now given by H 3 = H 2 + H kin = H + H g 34 H g = ig π Kφ φ K π ak 1 ig pβ 2 K ai a IKβ a Iβ a IK + 1 ak 2 pβ K + a Kβ + 1 b 2 qβ x + b β + 1 b β 2 x + b β q β β + π KM K b + b K M K π + 1ig b 2 β M K a KI b M K a KIβ MI q β 1 ig qβ M 2 I aik M K b β a IKβ M K b 1 2 qβ q β p β I + ig M I q β φ ig φ I q β M p Iβ + ig M K q Kβ φ I ig φ q Kβ MKI. 1 4 In the Hamiltonian description, the partial derivatives of the fields in 34 do not constitute canonical variables and must hence be regarded as x µ -dependent coefficients when setting up the canonical field equations. The relation of the canonical momenta p µν NM to the derivatives of the fields, a MNµ / x ν, is generally provided by the first canonical field equation 5. This means for the particular Hamiltonian 34 a MNµ x ν = H g p µν NM = 1 2 ig a MIµ a INν a MIν a INµ amnµ x ν + a MNν

18 18 Inhomogeneous local gauge transformations 1p 2 MNµν 1ig MIM q 2 Iµν φ N φ M q Iµν MIN, hence p Kµν = a Kν a Kµ x ν + ig a KIν a Iµ a KIµ a Iν M IK q Iµν φ + φ K q Iµν MI. 35 Rewriting Eq. 35 in the form p Kµν + ig M IK q Iµν φ igφ K q Iµν MI = a Kν = f Kµν, a Kµ x ν + ig a KIν a Iµ a KIµ a Iν we realise that the left-hand side transforms homogeneously according to Eq. 22. From Eq. 25, we already know that the same rule applies for the f µν. The canonical equation 35 is thus generally consistent with the canonical transformation rules. The corresponding reasoning applies for the canonical momenta q µν and q µν b Nµ x ν b Nµ x ν = H g q µν N bnµ x ν = H g q µν N bnµ = 1q 2 Nµν 1ig M 2 NI a IKµ M K b ν a IKν M K b µ + 12 ig M NI p Iµν + ig M KI q Kµν φ ig φ I q Kµν MK φ + b Nν = 1 2 q Nµν ig b ν M K a KIµ b µ M K a KIν MNI + b Nν 1 x ig φ µ 2 p Iµν + ig M K q Kµν φ I ig φ q Kµν MKI MNI, x ν hence with the canonical equation 35 q µν = b ν b µ x ν + ig M I a IKν M K b µ a IKµ M K b ν + ig M I aikν q µν = b ν x b µ µ x ν a IKµ x ν + ig a Iν a Kµ a Iµ a Kν ig b µ M K a KIν b ν M K a KIµ MI akiν ig φ K x a KIµ + ig a µ x ν Kν a Iµ a Kµ a Iν M I. 36 In order to check whether these canonical equations which are complex conjugate to each other are also compatible with the canonical transformation rules, we rewrite the first one concisely in matrix notation for the transformed fields MQ µν = MB ν MB µ + ig A x ν ν M B µ A µ M B ν Aν + ig x A µ µ x + ig A ν ν A µ A µ A ν Φ. Applying now the transformation rules for the gauge fields A ν, B µ from Eqs. 19, and for the base fields Φ from Eqs. 12, we find [ Mbν MQ µν = U x Mb µ + ig a µ x ν ν M b µ a µ M b ν φ K

19 Inhomogeneous local gauge transformations 19 aν + ig x a ] µ µ x + ig a ν ν a µ a µ a ν φ = UM q µν. The canonical equations 36 are thus compatible with the canonical transformation rules 25 provided that M T M = det M 2. Thus, the mass matrix M must be orthogonal. This restriction was already encountered with Eq. 33. We observe that both p Kµν and q µν, q µν occur to be skew-symmetric in the indices µ, ν. Here, this feature emerges from the canonical formalism and does not have to be postulated. Consequently, all products with the momenta in the Hamiltonian 34 that are symmetric in µ, ν must vanish. As these terms only contribute to the first canonical equations, we may omit them from H g if we simultaneously define p Kµν and q µν to be skew-symmetric in µ, ν. With regard to the ensuing canonical equations, the gauge Hamiltonian H g from Eq. 34 is then equivalent to H g = ig p µν K π β K φ φ K π β a Kβ ig p β I a IK a Kβ 1 q β 2 q β π β β K ig q M I a IK M K b β + b Kβ M K π β + ig a M I I q β p β I + ig M I q β φ ig φ I q β M p Iβ + ig M K q Kβ φ I ig φ q Kβ MKI! = p νµ K, qµν! = q νµ. 37 Setting the mass matrix M to zero, H g reduces to the gauge Hamiltonian of the homogeneous UN gauge theory [10]. The other terms describe the dynamics of the 4-vector gauge fields b. From the locally gauge-invariant Hamiltonian 34, the canonical equations for the base fields φ I, φ I are given by φ I = H 3 H3 π µ I φ I = H 3 H3 π µ I = H π µ I = H π µ I + ig a Iµ φ + M I b µ ig φ a Iµ + b µ M I. 38 These equations represent the generalised minimum coupling rules for our particular case of a system of two sets of gauge fields, a K and b. The canonical field equation from the b, b dependencies of H g follow as q µ K q µ = H g b Kµ = M K = H g b µ = π µ + ig a M I I q µ π µ K + ig qµ M I a IK M K. Inserting π, π as obtained from Eqs. 38 for a particular system Hamiltonian H, terms proportional to b I and b I emerge with no other dynamical variables involved. Such terms describe the masses of particles that are associated with the gauge fields b I.

20 20 Inhomogeneous local gauge transformations 4.3. Gauge-invariant agrangian As the system Hamiltonian H does not depend on the gauge fields a K and b, the gauge agrangian g that is equivalent to the gauge Hamiltonian H g from Eq. 34 is derived by means of the egendre transformation with p µν K p β a K K and, similarly q β b b qβ g = p β a K K + q β b x + b β qβ H g, from Eq. 35 and qµν, qµν from Eqs. 36. We thus have = 1 ak 2 pβ K a Kβ + 1 ak 2 pβ K x β = 1 2 pβ K p Kβ + 1 ak 2 pβ K + a Kβ 1 ig pβ 2 K = 1 2 q β ig q β q β = 1 2 q β 1 2 ig φ I a Kβ a KI a Iβ a KIβ a I M IK q Iβ φ + φ K q Iβ MI, q β 1 ig qβ 2 M I a IK M K b β a IKβ M K b M I p Iβ + ig M KI q Kβ φ ig φ I q Kβ MK φ x + b β β b q β + 1ig b 2 β M K a KI b M K a KIβ MI q β p Iβ + ig M KI q Kβ φ ig φ I q Kβ MK M q β b + b β q β. With the gauge Hamiltonian H g from Eq. 34, the gauge agrangian g is then g = 1 q β 2 q β π K ig a K φ + M K b + ig φ K a K b K M K π 1 p β 4 I + ig M I q β φ ig φ I q β M p Iβ + ig M K q Kβ φ I ig φ q Kβ MKI According to Eq. 23 and the relation for the canonical momenta p Iβ from Eq. 35, the last product can be rewritten as 1 4 f β I f Iβ, thus g = 1 4 f β I f Iβ 1 2 qβ q β π K ig a K φ + M K b + ig φ K a K b K M K π. With regard to canonical variables π K, π K, g is still a Hamiltonian. The final total gaugeinvariant agrangian 3 for the given system Hamiltonian H then emerges from the egendre transformation 3 = g + π φ + φ π Hφ I, φ I, π I, π I, x

21 Inhomogeneous local gauge transformations 21 = π φ x ig a φ Kφ K M K b K + x + ig φ Ka K b K M K π 1 4 f β I f Iβ 1 2 qβ q β H φ I, φ I, π I, π I, x. 39 As implied by the agrangian formalism, the dynamical variables are given by both the fields, φ I, φ I, a K, b, and b, and their respective partial derivatives with respect to the independent variables, x µ. Therefore, the momenta q and q of the Hamiltonian description are no longer dynamical variables in g but merely abbreviations for combinations of the agrangian dynamical variables, which are here given by Eqs. 36. The correlation of the momenta π I, π I of the base fields φ I, φ I to their derivatives are derived from the system Hamiltonian H via φ I x = H µ π µ + ig a Iµ φ + M I b µ I φ I x = H µ π µ ig φ a Iµ + b µ M I, 40 I which represents the minimal coupling rule for our particular system. Thus, for any globally gauge-invariant Hamiltonian Hφ I, π I, x, the amended agrangian 39 with Eqs. 40 describes in the agrangian formalism the associated physical system that is invariant under local gauge transformations Klein-Gordon system Hamiltonian As an example, we consider the generalised Klein-Gordon Hamiltonian [10] that describes an N-tuple of massless spin-0 fields H KG = π I π I. This Hamiltonian is clearly invariant under the inhomogeneous global gauge transformation 14. The reason for defining a massless system Hamiltonian H is that a mass term of the form φ I M I M K φ K that is contained in the general Klein-Gordon Hamiltonian is not invariant under the inhomogeneous gauge transformation from Eq. 14. According to Eqs. 39 and 40, the corresponding locally gauge-invariant agrangian 3,KG is then with f Kµν = a Kν q µν q µν π Iµ π Iµ 3,KG = π I π I 1 4 f β K f Kβ 1 2 q β q β, 41 a Kµ x ν + ig a KIν a Iµ a KIµ a Iν = b ν b µ x ν + ig M I aikν M K b µ a IKµ M K b ν + f IKµν φ K = b ν x b µ µ x ig b ν µ M K a KIν b ν M K a KIµ + φ K f KIµν MI = φ I ig a Iµφ M I b µ = φ I + ig φ a Iµ b µ M I.

22 22 Inhomogeneous local gauge transformations In matrix notation, the gauge-invariant agrangian 41 thus writes φ 3,KG = + ig φ a b φ M T x ig a φ Mb with f µν Mq µν q µν M T = Tr 1 4 f β f β 1 2 q β q β = a ν x a µ µ x + ig a ν ν a µ a µ a ν bν = M x b µ + ig µ x ν b ν x b µ µ x ν a ν Mb µ a µ Mb ν + f µν φ M T ig b µ M T a ν b ν M T a µ + φ f µν. The terms in parentheses in the first line of 3,KG can be regarded as the minimum coupling rule for the actual system. Under the inhomogeneous transformation prescription of the base fields from Eqs. 12 and the transformation rules of the gauge fields from Eqs. 19, the agrangian 3,KG is form-invariant. Moreover, the agrangian contains a term that is proportional to the square of the 4-vector gauge fields b b M T M b, which represents a Proca mass term for an N-tuple of possibly charged bosons. Setting up the Euler-agrange equation for the gauge fields b µ, we get q µ igm T a M T 1 q µ + M T φ ig a µ φ M T M b µ = 0. We observe that this equation describes an N-tuple massive bosonic fields b µ, in conjunction with their interactions with the massless gauge fields a Iµ and the base fields, φ I. Expanding the last term of the agrangian 41, we can separate this agrangian into a renormalisable r 3,KG part r 3,KG = πi π I 1f β 4 K f Kβ 1h β 2 h µν h β = b ν b µ x ν + ig M I a IKν M K b µ a IKµ M K b ν h µν = b ν x b µ µ x ig b ν µ M K a KIν b ν M K a KIµ MI, and into a non-renormalisable nr 3,KG part [ nr 3,KG = 1ig bβ 2 x b M I f β IK φ K φ K f β M bβ KI I x b ] g 2 [ + 1 b M 2 K a KIβ b β M K a KI f β I det M φ ] + φ K f β KI aiβ M b a I M b β + φk f β KI f Iβφ.

23 Inhomogeneous local gauge transformations 23 The first line vanishes if we restrict ourselves to real fields. nr 3,KG vanishes completely if g = 0, hence if all couplings to the massless gauge fields a IK are skipped. This corresponds to a pure shifting transformation that is generated by Eq. 13 with U = 1. For the case N = 1, hence for a single base field φ, the following twofold amended Klein-Gordon agrangian 3,KG φ 3,KG = + ig φ a m b φ x ig a φ m b 1f β f 4 β 1q β q 2 β is form-invariant under the combined local gauge transformation φ Φ = φ e iλ + ϕ, a µ A µ = a µ + 1 Λ g b µ B µ = b µ e iλ ig a µ + 1 Λ ϕ + 1 ϕ m g m x. µ The field tensors then simplify to f µν = a ν x a µ µ x ν q µν = b ν b µ x ν + ig a ν b µ a µ b ν + ig m q µν = b ν b µ x ν ig b µ a ν b ν a µ ig m φ aν x a µ µ x ν aν x a µ µ x ν With m 2 b b, this locally gauge-invariant agrangian contains a mass term for the complex bosonic 4-vector gauge field b µ. The subsequent equation for the massive gauge field b µ is thus q µ φ ig a q µ + m ig a µ φ m 2 b µ = 0. From the transformation rule for the fields, the rule for the momenta Q µν follows as Q µν = q µν e iλx. It is then easy to verify that the field equation is indeed form-invariant under the above combined local transformation of the fields φ, a µ, b µ. The agrangian 3,KG can again be split into a renormalisable part r 3,KG r 3,KG = f µν h µν φ + ig φ a m b = a ν x a µ µ x ν = b ν b µ x ν + ig a ν b µ a µ b ν h µν = b ν x b µ µ x ig b ν µ a ν b ν a µ and a non-renormalisable part nr nr 3,KG = ig m φ ig a φ m b 3,KG, h β φ φ h β ig m φφ f β φ. 1f β f 4 β 1h β h 2 β f β.

24 24 Inhomogeneous local gauge transformations 5. Conclusions With the present paper, we have worked out a complete non-abelian theory of inhomogeneous local gauge transformations. The theory was worked out as a canonical transformation in the realm of covariant Hamiltonian field theory. A particularly useful device was the definition of a gauge field matrix a I, with each matrix element representing a 4-vector gauge field. This way, the mutual interactions of base fields φ I and both sets of gauge fields, a I and b, attain a straightforward algebraic representation as ordinary matrix products. Not a single assumption or postulate needed to be incorporated in the course of the derivation. Moreover, no premise with respect to a particular potential energy function was required nor any draft on a symmetry breaking mechanism. The only restriction needed to render the theory consistent was to require the mass matrix to be orthogonal. Requiring a theory to be form-invariant under the SUN gauge group generally enforces all gauge fields to be massless. Yet, we are free to define other local gauge groups, under which we require the theory to be form-invariant. Defining a local shifting transformations of the base fields means to submit the given system to the action of fluctuating background fields. A local gauge invariance of the system s Hamiltonian then actually requires the existence of massive gauge fields. Specifically, the formalism enforces to introduce both a set of massless gauge fields and a set of massive gauge fields. The various mutual interactions of base and gauge fields that are described by the corresponding gauge-invariant agrangian 3 give rise to a variety of processes that can be used to test whether this beautiful formalism is actually reflected by nature. References [1] Weyl H 1919 Annalen der Physik IV Folge [2] Higgs P W 1964 Phys. Rev. ett [3] Kibble T W B 1967 Phys. Rev [4] Struckmeier Phys. G: Nucl. Part. Phys [5] Ryder 2006 Quantum Field Theory Cambridge University Press [6] Griffiths D 2008 Introduction to Elementary Particles. Wiley & Sons, New York [7] Cheng T P and i F 2000 Gauge Theory of elementary particle physics Oxford University Press, USA ISBN [8] De Donder T 1930 Théorie Invariantive Du Calcul des Variations Gaulthier-Villars & Cie., Paris [9] Weyl H 1935 Annals of Mathematics [10] Struckmeier and Reichau H 2013 General UN gauge transformations in the realm of covariant Hamiltonian field theory, in: Exciting Interdisciplinary Physics FIAS Interdisciplinary Science Series Springer, New York UR [11] Struckmeier 2009 Int.. Mod. Phys UR [12] osé and Saletan E 1998 Classical Dynamics Cambridge University Press, Cambridge