Covariant Hamiltonian Field Theory 3

Home , Hamiltonian field theory, Lagrange bracket, Legendre transformation

arXiv:0811.0508v6 [math-ph] 15 Dec 2020 eebr1,22 :8WP/NTUTO IEkfte FILE WSPC/INSTRUCTION 2:58 2020 16, December pc otesaetm oriae,i.e. coordinates, space-time the to spect eaiitcfil hoisadguetere r omnyformulate commonly density Lagrangian are a theories of gauge and theories field Relativistic Introduction 1. rmapsil xlctdpnec ntefu pc-iecoordina space-time four the on dependence explicit possible a from et hsi eetdb h atta h arnindensity Lagrangian the are that fact time re the and be by to space reflected description is of the This variables ensures rect. automatically independent which four footing, feat the equal characteristic that A is principle. proach action Hamilton’s the of from follow representation that equations field Euler-Lagrange the integrating h e ffields of set the qa otn ftesaetm oriae sabandoned is coordinates space-time the of physica footing given equal the in involved are that fields individual the enumerates hntetasto oaHmloindsrpini aei textbooks, in made is description Hamiltonian a to transition the When sa arnindsrpin ti rvdta oso bra the Poisson of that form proved is the It preserve description. that Lagrangian mappings usual defining are for transfor equations means canonical eral covariant field the canonical equations, field covariant Ha Lagrange covariant the of Whereas formulation sented. coordinate local consistent, A ASnmes 11.f 11.15Kc 11.10.Ef, numbers: PACS Keywords infinit equations. an field of f the function to general generating conforms most the the specify furthermore furnishes We i transformation particular, f In canonical fields examples. the imal various for of means rules by transformation demonstrated canonical derive under to invariant technique are The that exist 2-forms canonical OAIN AITNA IL THEORY FIELD HAMILTONIAN COVARIANT il hoy aitna est;covariant. density; Hamiltonian theory; Field : J a-o-aeSr ,648Fakuta an Germany Main, am Frankfurt 60438 1, Max-von-Laue-Str. φ oanWlgn oteUiesta rnfr mMain am Goethe-Universität Frankfurt Wolfgang Johann RE TUKEE n NRA REDELBACH ANDREAS and STRUCKMEIER URGEN S emotznrmfu cwroefrcugGmbH für Schwerionenforschung Helmholtzzentrum GSI ¨ I n vnyo l orderivatives four all on evenly and lnkt.1 49 amtd,Germany Darmstadt, 64291 1, Planckstr. L 1 , 2 , 3 , 4 eie 4Dcme 2020 December 14 Revised h pc-ieeouino h ed sotie by obtained is fields the of evolution space-time The . eevd1 uy2007 July 18 Received [email protected] and L 1 = L ( φ I ,∂ ∂ ∂φ ainter ffr oegen- more offers theory mation kt,Lgag rces and brackets, Lagrange ckets, itna edter spre- is theory field miltonian rnfrain ftefields. the of transformations I ssonta h infinites- the that shown is t ∂ ∂ ∂φ x , o eeaigfntosis functions generating rom smlsaetm tpthat step space-time esimal r fNehrstheorem. Noether’s of orm qiaett h Euler- the to equivalent µ I edeutosta the than equations field .Hri,teidx“ index the Herein, ). 5 ftoefilswt re- with fields those of , 1 , 2 nteepresenta- these In . L eed apart — depends four-dimensional aiitcycor- lativisticly r fti ap- this of ure ntebasis the on d tes system. l rae on treated x µ on — the I ” December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

2 J. Struckmeier, A. Redelbach

tions, the Hamiltonian density H is defined to depend on the set of scalar fields φI and on one set of conjugate scalar fields πI that counterpart the time derivatives of the φI . Keeping the dependencies on the three spatial derivatives ∂ν φI of the fields φI , the functional dependence of the Hamiltonian is then defined as µ H = H(φI ,ππI , ∂ν φI , x ). The canonical field equations then emerge as time derivatives of the scalar fields φI and πI . In other words, the time variable is singled out of the set of independent space-time variables. While this formulation is doubtlessly valid and obviously works for the purpose pursued in these presentations, it closes the door to a full-fledged Hamiltonian field theory. In particular, it appears to be impossible to formulate a theory of canonical transformations on the basis of this particular definition of a Hamiltonian density. On the other hand, numerous papers were published that formulatea covariant Hamiltonian description of field theories where — similar to the Lagrangian formalism — the four independent variables of space-time are treated equally. These papers are generally based on the pioneering works of De Donder6 and Weyl7. The key point of this approach is that the Hamiltonian density H is now defined to de- µ pend on a set of conjugate 4-vector fields πI that counterbalance the four derivatives µ µ ∂µφI of the Lagrangian density L, so that H = H(φI , πI , x ). Corresponding to the Euler-Lagrange equations of field theory, the canonical field equations then take on a symmetric form with respect to the four independent variables of space-time. This approach is commonly referred to as “multisymplectic” or “polysymplectic field theory”, thereby labeling the covariant extension of the symplectic geometry of the conventional Hamiltonian theory8,9,10,11,12,13,14,15. Mathematically, the phase space of multisymplectic Hamiltonian field theory is defined within modern differential geometry in the language of “jet bundles”16,17. Obviously, this theory has not yet found its way into mainstream textbooks. One reason for this is that the differential geometry approach to covariant Hamiltonian field theory is far from being straightforward and raises mathematical issues that are not yet clarified (see, for instance, the discussion in Ref. 12). Furthermore, the approach is obviously not unique — there exist various options to define geometric objects such as Poisson brackets8,11,14. As a consequence, any discussion of the matter is unavoidably shifted into the realm of mathematics. With the present paper, we do not pursue the differential geometry path but provide a local coordinate treatise of De Donder and Weyl’s covariant Hamiltonian field theory. The local description enables us to keep the mathematics on the level of tensor calculus. Nevertheless, the description is chart-independent and thus applies to all local coordinate systems. With this property, our description is sufficiently general from the point of view of physics. Similar to textbooks on Lagrangian gauge theories, we maintain a close tie to physics throughout the paper. Our paper is organized as follows. In Sec. 2, we give a brief review of De Donder and Weyl’s approach to Hamiltonian field theory in order to render the paper self-contained and to clarify notation. After reviewing the covariant canonical field equations, we evince the Hamiltonian density H to represent the eigenvalue of the December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 3

energy-momentum tensor and discuss the non-uniqueness of the field vector πI . The main benefit of the covariant Hamiltonian approach is that it enables us to formulate a consistent theory of covariant canonical transformations. This is demonstrated in Sec. 3. Strictly imitating the point mechanics’ approach18,19, we set up the transformation rules on the basis of a generating function by requiring the variational principle to be maintained20. In contrast to point mechanics, the generating µ function F1 now emerges in our approach as a 4-vector function. We recover a characteristic feature of canonical transformations by deriving the symmetry relations of the mutual partial derivatives of original and transformed fields. By means of covariant Legendre transformations, we show that equivalent transformation rules µ µ µ are obtained from generating functions F2 , F3 , and F4 . Very importantly, each of these generating functions gives rise to a specific set of symmetry relations of original and transformed fields. The symmetry relations set the stage for proving that 4-vectors of Poisson and Lagrange brackets exist that are invariant with respect to canonical transformations of the fields. We furthermore show that each vector component of our definition of a (1, 2)-tensor, i.e. a “4-vector of 2-forms ωµ” is invariant under canonical transformations — which establishes Liouville’s theorem of canonical field theory. We conclude this section deriving the field theory versions of the Jacobi identity, Pois- son’s theorem, and the Hamilton-Jacobi equation. Similar to point mechanics, the action function S of the Hamilton-Jacobi equation is shown to represent a generating function S ≡ F 2 that is associated with the particular canonical transformation that maps the given Hamiltonian into an identically vanishing Hamiltonian. In Sec. 4, examples of Hamiltonian densities are reviewed and their pertaining field equations are derived. As the relativistic invariance of the resulting fields equations is ensured if the Hamiltonian density H is a Lorentz scalar, various equations of relativistic quantum field theory are demonstrated to embody, in fact, canonical field equations. In particular, the Hamiltonian density engendering the Klein-Gordon equation manifests itself as the covariant field theory analog of the harmonic oscillator Hamiltonian of point mechanics. Section 5 starts sketching simple examples of canonical transformations of Hamiltonian systems. Similar to the case of classical point mechanics, the main advantage of the Hamiltonian over the Lagrangian description is that the canonical transformation approach is not restricted to the class of point transformations, i.e., to cases where the transformed fields ΦI only depend on the original fields φI . The most general formulation of Noether’s theorem is, therefore, obtained from a general infinitesimal canonical transformation. As an application of this theorem, we show that an invariance with respect to a shift in a space-time coordinate leads to a corresponding conserved current that is given by the pertaining column vector of the energy-momentum tensor. By specifying its generating function, we furthermore show that an infinitesimal step in space-time which conforms to the canonical field equations itself establishes a canonical transformation. Similar to the corresponding time-step transformation December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

4 J. Struckmeier, A. Redelbach

of point mechanics, the generating function is mainly determined by the system’s Hamiltonian density. It is precisely this canonical transformation which ensures that a Hamiltonian systems remains a Hamiltonian system in the course of its space- time evolution. The existence of this canonical transformation is thus crucial for the entire approach to be consistent. As canonical transformations establish mappings of one physical system into another, canonically equivalent system, it is remarkable that Higgs’ mechanism of spontaneous symmetry breaking can be formulated in terms of a canonical transformation. This is shown in Sec. 5.11. We close our treatise with a discussion of the generating function of a non-Abelian gauge transformation.

2. Covariant Hamiltonian density 2.1. 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. 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 function φ(x) that is uniquely given at a spatial location x1, x2, x3 at time t ≡ x0. The differentiable function φ(x) is called a primary field. In this notation, the index µ runs from 0 to 3, hence distinguishes the four independent variables of space-time xµ ≡ (x0, x1, x2, x3) ≡ (ct,x,y,z), and xµ ≡ (x0, x1, x2, x3) ≡ (ct, −x, −y, −z). We furthermore assume that the given physical problem can be described in terms of I =1,...,N — possibly interacting — scalar fields φI (x), with the index “I” enumerating the individual fields. In order to clearly distinguish scalar quantities from vector quantities, we denote the latter with boldface letters. Throughout our paper, the summation convention is used. This means that whenever a pair of the same upper and lower indices appears on one side of an equation, this index is to be summed over. If 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 Lagrangian description of the dynamics of a continuous system is based on the Lagrangian density function L that is supposed to carry the complete information on the given physical system. In a first-order field theory, the Lagrangian density L is defined to depend on the φI , possibly on the vector of independent variables x, and on the four first derivatives of the fields φI with respect to the independent variables, i.e., on the 1-forms

∂∂φI ≡ (∂ctφI , ∂xφI , ∂yφI , ∂zφI ).

The Euler-Lagrange ﬁeld equations are then obtained as the zero of the variation December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 5

δS of the action integral

S = L(φI ,∂∂∂φI , x) dx (1) Z as ∂ ∂L ∂L α − =0. (2) ∂x ∂(∂αφI ) ∂φI To derive the equivalent covariant Hamiltonian description of continuum dynamics, we first define for each primary field φI (x) a 4-vector of conjugate momentum fields µ πI (x). Its components are given by

µ ∂L ∂L πI = ≡ . (3) ∂(∂µφI ) ∂φI ∂ ∂xµ The 4-vector πI is thus induced by the Lagrangian L as the dual counterpart of the 1-form ∂∂φI . For the entire set of N scalar fields φI (x), this establishes a set of N conjugate 4-vector fields. With this definition of the 4-vectors of canonical momenta πI (x), we can now define the Hamiltonian density H(φI ,ππI , x) as the covariant Legendre transform of the Lagrangian density L(φI ,∂∂∂φI , x) ∂φ H(φ ,ππ , x)= πα J − L(φ ,∂∂∂φ , x). (4) I I J ∂xα I I At this point we suppose that L is regular, hence that for each index “I” the Hesse 2 matrices (∂ L/∂(∂µφI ) ∂(∂ν φI )) are non-singular. This ensures that H takes over the complete information on the given dynamical system from L by means of the Legendre transformation. The definition of H by Eq. (4) is referred to in literature as the “De Donder-Weyl” Hamiltonian density. µ Obviously, the dependencies of H and L on the φI and the x only differ by a sign, ∂H ∂L ∂H ∂L = − , = − . ∂φ ∂φ ∂xµ ∂xµ I I expl expl

These variables do not take part in the Legendre transformation of Eqs. (3), (4). With regard to this transformation, the Hamiltonian density H is, therefore, to µ be considered as a function of the πI only, and, correspondingly, the Lagrangian density L as a function of the ∂µφI only. In order to derive the canonical ﬁeld µ equations, we calculate from Eq. (4) the partial derivative of H with respect to πI ,

∂H α ∂φJ ∂φI µ = δIJ δµ α = µ . ∂πI ∂x ∂x µ According to the definition of πI in Eq. (3), the second and the third terms on the right hand side cancel. In conjunction with the Euler-Lagrange equation, we obtain the set of covariant canonical field equations finally as α ∂H ∂φI ∂H ∂πI µ = µ , = − α . (5) ∂πI ∂x ∂φI ∂x December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

6 J. Struckmeier, A. Redelbach

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 Lagrangian formulation of Eq. (2). Provided that the Lagrangian density L is a Lorentz scalar, the dynamics of the fields is invariant with respect to Lorentz transformations. The covariant Legendre 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 L — and hence H — represents a Lorentz scalar.

2.2. Energy-Momentum Tensor ν In the Lagrangian description, the canonical energy-momentum tensor θµ is de- ﬁned as the following mixed second rank tensor

ν ∂L ∂φI ν θµ = µ − δµ L. (6) ∂(∂ν φI ) ∂x µ With the definition (3) of the conjugate momentum fields πI , and the Hamiltonian density of Eq. (4), the energy-momentum tensor (6) is equivalently expressed as ∂φ ∂φ θ ν = πν I + δν H− πα I . (7) µ I ∂xµ µ I ∂xα If the Hamiltonian describes the dynamics of a single field, the inner product of the ν µ mixed tensors θµ of Eq. (7) with the (1, 1) tensor ∂ν φ π yields ✘✘ ✘ ∂φ ∂φ ✘∂φ✘ ∂φ ∂φ ✘∂φ✘✘ θ β πα = πβ✘✘ πα + δβ H πα − πα✘✘ πβ , α ∂xβ ✘ ∂xα ∂xβ α ∂xβ ✘ ∂xα ∂xβ hence ∂φ θ β − H δβ πα =0. α α ∂xβ This shows that the value of the De Donder-Weyl Hamiltonian density H consti- ν tutes the eigenvalue of the energy-momentum tensor θµ with the (1, 1) eigenten- ν sor ∂µφ π . By identifying H as the eigenvalue of the energy-momentum tensor, we obtain a clear interpretation of the physical meaning of the De Donder-Weyl Hamiltonian density H. An important property of the energy-momentum tensor is revealed by calculat- α α ing the divergence ∂θµ /∂x . From the definition (7), we find

∂θ α ∂H ∂φ ∂H ∂πβ ∂H ∂πα ∂φ µ = δα I + I + + I I ∂xα µ ∂φ ∂xα β ∂xα ∂xα ∂xα ∂xµ I ∂πI expl!

2 β 2 α ∂ φI α ∂πI ∂φI β ∂ φI + πI µ α − δµ α β + πI α β ∂x ∂x ∂x ∂x ∂x ∂x ! December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 7

Inserting the canonical ﬁeld equations (5), this becomes ∂θ α ∂H µ = . (8) ∂xα ∂xµ expl ν We observe that the Hamiltonian H — through its φI and explicit x dependencies j j j j — only determines the divergences ∂πI /∂x and ∂θν /∂x of both the canonical momentum tensor and the energy-momentum tensor, but not the individual com- ν ν ponents πI and θµ . If the Hamiltonian density H does not explicitly depend on the independent variable xµ, then H is obviously invariant with respect to a shift of the reference system along the xµ axis. Then, the components of the µ-th column of the energy- momentum tensor satisfy the continuity equation ∂θ α ∂H µ =0 ⇐⇒ =0. ∂xα ∂xµ expl

Using the deﬁnition (7) of the energy-momentum tensor, we infer from Eq. (8) ∂θ α ∂θ α µ = µ . ∂xα ∂xα expl

Based on the four independent variables xµ of space-time, this divergence rela-

tion for the energy-momentum tensor constitutes the counterpart to the relation dH/dt = ∂H/∂t of the time derivatives of the Hamiltonian function of point mechanics. Yet, such a relation does not exist in general for the Hamiltonian density H of ﬁeld theory. As we easily convince ourselves, the derivative of H with respect to xµ is not uniquely determined by its explicit dependence on xµ ∂H ∂H ∂H ∂πα ∂H ∂φ = + I + I ∂xµ ∂xµ ∂πα ∂xµ ∂φ ∂xµ expl I I

∂H ∂φ ∂πα ∂φ ∂πβ = + I I − I I ∂xµ ∂xα ∂xµ ∂xµ ∂xβ expl

∂H ∂φ ∂πν ∂πβ = + k α I , k ν = I − δν I . (9) ∂xµ Iµ ∂xα Iµ ∂xµ µ ∂xβ expl

Owing to the fact that the number of independent variables is greater than one,

the two rightmost terms of Eq. (9) constitute a sum. In contrast to the case of point mechanics, these terms generally do not cancel by virtue of the canonical ﬁeld equations.

2.3. Non-uniqueness of the conjugate vector fields πI From the right hand side of the second canonical field equation (5) we observe that the dependence of the Hamiltonian density H on φI only determines the divergence of the conjugate vector field πI . Vice-versa, the canonical field equations are invari- µ ν ant with regard to all transformations of the mixed tensor (∂πI /∂x ) that preserve December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

8 J. Struckmeier, A. Redelbach

its trace. The expression ∂H/∂φI thus only quantifies the change of the flux of πI through an infinitesimal space-time volume around a space-time location x. The vector field πI itself is, therefore, only determined up to a vector field ηI (x) that leaves its divergence invariant ∂ηα πµ 7→ π˜µ = πµ − ηµ, I =0. (10) I I I I ∂xα With this condition fulfilled, we are allowed to subtract a field ηI (x) from πI (x) without changing the canonical field equations (5), hence the description of the dynamics of the given system. In the Lagrangian formalism, the transition (10) corresponds to the transformation ∂φ (x) L 7→ L˜ = L− ηα(x) I . I ∂xα which leaves — under the condition (10) — the Euler-Lagrange equations (2) invariant but obviously not the value of the Lagrangian. The Hamiltonian density H˜, expressed as a function of π˜I , is obtained from the Legendre transformation ˜ µ ∂L µ µ π˜I = = πI − ηI ∂(∂µφI ) ∂φ H˜(φ ,π˜ , x)=˜πα I − L˜(φ ,∂∂∂φ , x) I I I ∂xα I I ∂φ ∂φ ∂φ = πα I − ηα I − L + ηα I I ∂xα I ∂xα I ∂xα = H(φI ,ππI , x). In contrast to the Lagrangian density, the value of the Hamiltonian density H thus remains invariant under the action of the shifting transformation (10). This means for the canonical field equations (5) ˜ ˜ α ˜ ∂H ∂H ∂φI ∂H ∂H ∂πI ∂H ∂H µ = µ = µ , = = − α , µ = µ . ∂π˜ ∂π ∂x ∂φI ∂φI ∂x ∂x ∂x I I expl expl

Thus, both momentum ﬁelds π˜I and πI equivalently describe the same physical α α system. In other words, we can switch from πI to π˜I = πI − ηI with ∂ηI /∂x =0 without changing the description of the given physical system.

3. Canonical transformations in covariant Hamiltonian ﬁeld theory

3.1. Generating functions of type F 1(φφ, Φ,x) Similar to the canonical formalism of point mechanics, we call a transformation of the ﬁelds (φφ,ππ) 7→ (ΦΦ,ΠΠ) canonical if the form of the variational principle that is based on the action integral (1) is maintained, ∂φ ∂Φ δ πα I − H(φφ,πππ, x) dx = δ Πα I − H′(ΦΦ,Π, x) dx. (11) I ∂xα I ∂xα ZR ZR December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 9

Equation (11) tells us that the integrands may differ by the divergence of a vector µ field F1 , whose variation vanishes on the boundary ∂R of the integration region R within space-time ∂F α δ 1 dx = δ F αdS =0! . ∂xα 1 α ZR I∂R The immediate consequence of the form invariance of the variational principle is the form invariance of the covariant canonical field equations (5) ′ ′ α ∂H ∂ΦI ∂H ∂ΠI µ = µ , = − α . (12) ∂ΠI ∂x ∂ΦI ∂x For the integrands of Eq. (11) — hence for the Lagrangian densities L and L′ — we thus obtain the condition ∂F α L = L′ + 1 (13) ∂xα ∂φ ∂Φ ∂F α πα I − H(φφ,πππ, x) = Πα I − H′(ΦΦ,Π, x)+ 1 . (14) I ∂xα I ∂xα ∂xα µ µ With the definition F1 ≡ F1 (φφ,Φ, x), we restrict ourselves to a function of exactly those arguments that now enter into transformation rules for the transition from µ the original to the new fields. The divergence of F1 writes, explicitly, ∂F α ∂F α ∂φ ∂F α ∂Φ ∂F α 1 = 1 I + 1 I + 1 . (15) ∂xα ∂φ ∂xα ∂Φ ∂xα ∂xα I I expl

The rightmost term denotes the sum over the explicit dependence of the generating µ ν function F1 on the x . Comparing the coefficients of Eqs. (14) and (15), we find the local coordinate representation of the field transformation rules that are induced µ by the generating function F1 ∂F µ ∂F µ ∂F α πµ = 1 , Πµ = − 1 , H′ = H + 1 . (16) I ∂φ I ∂Φ ∂xα I I expl

The transformation rule for the Hamiltonian density implies that summation over α

is to be performed. In contrast to the transformation rule for the Lagrangian density L of Eq. (14), the rule for the Hamiltonian density is determined by the explicit µ µ dependence of the generating function F1 on the x . Hence, if a generating function does not explicitly depend on the independent variables, xµ, then the value of the Hamiltonian density is not changed under the particular canonical transformation emerging thereof. µ Swapping the sequence of calculating the derivatives of F1 (φ, Φ, x) with respect to the fields in the argument list of the generating function, we find a symmetry relation between original and transformed fields ∂ ∂F µ ∂ ∂F µ ∂πµ ∂Πµ 1 = 1 ⇒ I = − J . (17) ∂Φ ∂φ ∂φ ∂Φ ∂Φ ∂φ J I I J J I The emerging of symmetry relations is a characteristic feature of canonical transformations, hence transformations of the fields that follow from generating functions. December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

10 J. Struckmeier, A. Redelbach

Consequently, Eq. (17) does not apply for arbitrary ﬁeld transformations that cannot be derived from a generating function. To derive the transformation rule for the energy-momentum tensor (7), we express the tensor in terms of the transformed coordinates, subsequently apply the transformation rules (16), and insert the divergence expression from Eq. (15) ∂Φ ∂Φ Θ ν = δν H′ + πν I − δν πα I µ µ I ∂xµ µ I ∂xα ∂F α ∂F ν ∂Φ ∂F α ∂Φ = δν H + δν 1 − 1 I + δν 1 I µ µ ∂xα ∂Φ ∂xµ µ ∂Φ ∂xα expl I I

∂F α ∂F ν ∂F ν ∂F ν ∂φ = δν H + δν 1 − 1 − 1 − 1 I µ µ ∂xα ∂xµ ∂xµ ∂φ ∂xµ expl expl I !

∂F α ∂F α ∂F α ∂φ + δν 1 − 1 − 1 I µ ∂xα ∂xα ∂φ ∂xα expl I !

∂φ ∂φ ∂F ν ∂F ν ∂F α = δν H + πν I − δν πα I + 1 − 1 + δν 1 µ I ∂xµ µ I ∂xα ∂xµ ∂xµ µ ∂xα expl ν να ∂F ∂ ∂Kµ = θ ν + 1 − k ν (x), k ν = δαF ν − δν F α = , µ ∂xµ µ µ ∂xα µ 1 µ 1 ∂xα expl with K να skew-symmetric in the upper indices, ν, α. The divergences ∂k β/∂xβ µ µ vanish identically, ∂k β ∂2K βα ∂2F β ∂2F α µ = µ = 1 − δβ 1 ≡ 0. ∂xβ ∂xα∂xβ ∂xµ∂xβ µ ∂xα∂xβ As already discussed in Sect. 2.2, we have seen in Eq. (8) that the energy-momentum ν ν tensor θµ is determined only up to divergence-free functions. Therefore, the kµ terms can be skipped from the transformation rule for the energy-momentum tensor elements, yielding ∂F µ Θ µ = θ µ + 1 . ν ν ∂xν expl

3.2. Generating functions of type F 2(φφ, Π ,x) The generating function of a canonical transformation can alternatively be expressed in terms of a function of the original fields φI and of the new conjugate µ fields ΠI . To derive the pertaining transformation rules, we perform the covariant Legendre transformation µ µ µ µ µ ∂F1 F2 (φφ,Π, x)= F1 (φφ,Φ, x)+ΦJ ΠJ , ΠI = − . (18) ∂ΦI µ µ µ By definition, the functions F1 and F2 agree with respect to their φI and x dependencies ∂F µ ∂F µ ∂F α ∂F α 2 = 1 = πµ, 2 = 1 = H′ − H. ∂φ ∂φ I ∂xα ∂xα I I expl expl

December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 11

µ The variables φI and x do not take part in the Legendre transformation from µ Eq. (18). Therefore, the two F2 -related transformation rules coincide with the µ µ µ respective rules derived previously from F1 . As F1 does not depend on the ΠI µ whereas F2 does not depend on the ΦI , the new transformation rule thus follows µ ν from the derivative of F2 with respect to ΠJ , µ µ ∂F2 ∂ΠJ µ ν =ΦJ ν =ΦJ δIJ δν . ∂ΠI ∂ΠI We thus end up with set of transformation rules ∂F µ ∂F µ ∂F α πµ = 2 , Φ δµ = 2 , H′ = H + 2 , (19) I ∂φ I ν ∂Πν ∂xα I I expl

which is equivalent to the set (16) by virtue of the Legendre transformation (18) 2 µ if the matrices (∂ F1 /∂φI ∂ΦJ ) are non-singular for all indices “µ”. Writing the second rule in explicit form

0 3 µ ∂F2 ∂F2 ∂F2 0 = ... = 3 =ΦI , ν =0 for ν 6= µ, ∂ΠI ∂ΠI ∂ΠI

we conclude that the µ-component of the generating vector function F 2 must depend on the µ-component of the transformed momentum field vector ΠI but cannot µ µ µ depend on the ν 6= µ components of ΠI , hence F2 = F2 (φ, Π , x) for all indices µ =0,..., 3. µ From the second partial derivations of F2 one immediately derives the symmetry relation ∂ ∂F µ ∂ ∂F µ ∂πµ ∂Φ 2 = 2 ⇒ I = J δµ, (20) ∂Πν ∂φ ∂φ ∂Πν ∂Πν ∂φ ν J I I J J I which exhibits the restriction that the µ-component of the original momentum field vector πI can only depend on the µ-component of the transformed momentum field vector ΠJ .

3.3. Generating functions of type F 3(ΦΦ,πππ,x) By means of the Legendre transformation

µ µ µ µ µ ∂F1 F3 (ΦΦ,πππ, x)= F1 (φφ,Φ, x) − φJ πJ , πI = , (21) ∂φI the generating function of a canonical transformation can be converted into a func- µ µ tion of the new ﬁelds ΦI and the original conjugate ﬁelds πI . The functions F1 and µ µ F3 agree in their dependencies on ΦI and x , ∂F µ ∂F µ ∂F α ∂F α 3 = 1 = −Πµ, 3 = 1 = H′ − H. ∂Φ ∂Φ I ∂xα ∂xα I I expl expl

December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

12 J. Struckmeier, A. Redelbach

Consequently, the pertaining transformation rules agree with those of Eq. (16). The µ ν new rule follows from the dependence of F3 on the πI : µ µ ∂F3 ∂πJ µ µ ν = −φJ ν = −φJ δIJ δν = −φI δν . ∂πI ∂πI 2 µ For (∂ F1 /∂φI ∂ΦJ ) non-singular, we thus get a third set of equivalent transformation rules, ∂F µ ∂F µ ∂F α Πµ = − 3 , φ δµ = − 3 , H′ = H + 3 . (22) I ∂Φ I ν ∂πν ∂xα I I expl

The second rule shows that the µ-component of the generating vector function F 3 depends and only depends on the µ-component of the momentum field vector πI , as 0 3 µ ∂F3 ∂F3 ∂F3 0 = ... = 3 = −φI , ν =0 for ν 6= µ, ∂πI ∂πI ∂πI µ µ µ hence F3 = F3 (Φ, π , x) for all indices µ =0,..., 3. The pertaining symmetry relation between original and transformed fields µ emerging from F3 follows as ∂ ∂F µ ∂ ∂F µ ∂Πµ ∂φ 3 = 3 ⇒ I = J δµ. (23) ∂πν ∂Φ ∂Φ ∂πν ∂πν ∂Φ ν J I I J J I Similarly to (20), Eq. (23) shows that the µ-component of the transformed momentum field vector ΠI can only depend on the µ-component of the original momentum field vector πJ .

3.4. Generating functions of type F 4(ΠΠ,πππ,x) Finally, by means of the Legendre transformation µ µ µ µ µ ∂F3 F4 (ΠΠ,πππ, x)= F3 (ΦΦ,πππ, x)+ΦJ ΠJ , ΠI = − (24) ∂ΦI we may express the generating function of a canonical transformation as a function µ µ µ of both the original and the transformed conjugate ﬁelds πI , ΠI . The functions F4 µ µ µ and F3 agree in their dependencies on the πI and x , ∂F µ ∂F µ ∂F α ∂F α 4 = 3 = −φ δµ , 4 = 3 = H′ − H. ∂πν ∂πν I ν ∂xα ∂xα I I expl expl

The related pair of transformation rules thus corresponds to that of Eq. (22). The µ ν new rule follows from the dependence of F4 on the ΠJ , µ µ ∂F4 ∂ΠJ µ µ ν =ΦJ ν =ΦJ δIJ δν =ΦI δν . ∂ΠI ∂ΠI December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 13

2 µ ν Under the condition that (∂ F3 /∂ΦI ∂πJ ) are non-singular, we thus get a fourth set of equivalent transformation rules ∂F µ ∂F µ ∂F α Φ δµ = 4 , φ δµ = − 4 , H′ = H + 4 . (25) I ν ∂Πν I ν ∂πν ∂xα I I expl

Writing the rules (25) explicitly, we observe that the µ-component of a generating

vector function of type F 4 must depend on the µ-components of the momentum field vectors πI and ΠI as the fields ΦI and φI do not generally vanish 0 3 0 3 ∂F4 ∂F4 ∂F4 ∂F4 0 = ... = 3 =ΦI , 0 = ... = 3 = −φI . ∂ΠI ∂ΠI ∂πI ∂πI µ Yet, F4 cannot depend on the other momentum vector components µ µ ∂F4 ∂F4 ν =0, ν =0 for ν 6= µ, ∂ΠI ∂πI which means that 0 0 0 0 3 3 3 3 F4 = F4 Π , π , x ,...,F4 = F4 Π , π , x . The symmetry relations between original and transformed fields follow again from swapping the sequence of second derivatives of the respective generating function. For a generating function of type F 4, one concludes ∂ ∂F µ ∂ ∂F µ ∂Φ ∂φ 4 = 4 ⇒ I δµ = − J δµ. (26) ∂πα β β ∂πα ∂πα β β α J ∂ΠI ! ∂ΠI J J ∂ΠI The tensor equation (26) allows for two contractions. For µ = α, one finds

∂ΦI ∂φJ β = −4 β , ∂πJ ∂ΠI whereas the contraction µ = β yields

∂ΦI ∂φJ 4 α = − α . ∂πJ ∂ΠI The contracted equations are simultaneously satisﬁed if and only if

∂ΦI ∂φJ µ =0, µ =0, (27) ∂πJ ∂ΠI

which means that the transformed fields ΦI can only depend on the original fields µ φJ but not on the original momentum fields πJ . Correspondingly, the original fields µ φJ cannot depend on the transformed momentum fields ΠI . We conclude that — in contrast to classical point dynamics — only point transformations are allowed in the realm of classical field theory in order for a transformation to be canonical. The particular subgroup of canonical transformations whose transformed mo- µ mentum fields ΠJ do not depend on the original fields φI is referred to as the group of momentum point transformations. The respective derivatives ∂Πµ ∂πµ J =0, I = 0 (28) ∂φI ∂ΦJ December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

14 J. Struckmeier, A. Redelbach

then satisfy trivially the symmetry condition (17). Equations (28) are suﬃcient to warrant the invariance of Poisson brackets under canonical transformations. This will be discussed in Sect. 3.7. In summary, the generating functions of canonical transformations in the realm of classical ﬁelds are given by functions with argument lists of the form

µ µ µ µ µ F1 = F1 φI , ΦI , x , F2 = F2 φI , ΠI , x µ µ µ µ µ µ µ F3 = F3 πI , ΦI , x, F4 = F4 πI , ΠI , x. µ µ The πI and ΠI in the argument lists of the µ-component of the respective generating vector function indicate a dependence on merely the µ-component of the momentum ﬁeld vectors rather than a dependence on the entire vectors πI and ΠI . This is necessary for the generating functions to be equivalent, hence to encode the same information on the transformation of the ﬁelds.

3.5. Consistency check of the canonical transformation rules As a test of consistency of the canonical transformation rules derived in the preced- µ ing four sections, the rules obtained from the generating function F1 are recovered µ from a Legendre transformation of F4 . Both generating functions are related by µ µ µ µ µ µ µ ∂F4 µ ∂F4 F1 (φφ,Φ, x)= F4 (ππ,Π, x)+ φJ πJ − ΦJ ΠJ , ΦI δν = ν , φI δν = − ν . ∂ΠI ∂πI µ µ In this case, the generating functions F1 and F4 only agree in their explicit dependence on xµ. This involves the common transformation rule ∂F α ∂F α 1 = 4 = H′ − H. ∂xα ∂xα expl expl

µ µ In the actual case, we thus transform at once two ﬁeld variables φI , ΦI and πI , ΠI . µ The transformation rules associated with F1 follow from its dependencies on both φI and ΦI according to µ µ µ ✟✟α µ α ∂F1 ∂F1 ∂ΦJ ∂F4 ✟∂πJ ∂F4 ∂ΠJ + = ✟α + α ∂φI ∂ΦJ ∂φI ✟∂πJ ∂φI ∂ΠJ ∂φI ✚µ µ µ ∂π✚J µ ∂ΦJ ∂ΠJ + πI + φJ✚ − ΠJ − ΦJ ✚ ∂φI ∂φI ∂φI ✟✟α ✚µ ✟µ ∂ΠJ µ µ ∂ΦJ ∂✚ΠJ = ΦJ✟δα + πI − ΠJ − ΦJ✚ ✟ ∂φI ∂φI ✚ ∂φI µ µ ∂ΦJ = πI − ΠJ . ∂φI Comparing the coeﬃcients on the left- and right-hand sides, we encounter the transformation rules µ µ µ ∂F1 µ ∂F1 πI = , ΠI = − . ∂φI ∂ΦI December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 15

As expected, the rules obtained previously in Eq. (16) are recovered. The same µ result follows if we differentiate F1 with respect to ΦI . Another consistency test is to derive the transformation rules (25) on the basis of an equivalent form of the action functional. Because of ∂φ ∂ ∂πα πα I = (παφ ) − I φ , (29) I ∂xα ∂xα I I ∂xα I the condition (11) for the form-invariance of the action principle can be expressed equivalently as ∂πα ∂Πα δ φ I + H(φ ,ππ , x) d4x =! δ Φ I + H′(Φ ,Π , x) d4x. I ∂xα I I I ∂xα I I ZR ZR Since the variation of the fields vanishes by assumption on the boundary ∂R of the integration region R, the total divergence in (29) does not contribute to the variation of the action functional. Again, the integrands may differ by the divergence of a vector field F4 ∂Πα ∂πα ∂F α Φ I + H′(Φ ,Π , x)= φ I + H(φ ,ππ , x)+ 4 . I ∂xα I I I ∂xα I I ∂xα With F4 = F4(πI ,ΠI , x), the explicit form of the integrand condition is ∂Πβ ∂πβ Φ δα I + H′(Φ ,Π , x)= φ δα I + H(φ ,ππ , x) I β ∂xα I I I β ∂xα I I ∂F α ∂πβ ∂F α ∂Πβ ∂F α + 4 I + 4 I + 4 . β ∂xα β ∂xα ∂xα ∂πI ∂ΠI expl

Comparing the coeﬃcients of the momentum ﬁeld derivatives now yield s the trans-

formation rules ∂F α ∂F α ∂F α Φ δα = 4 , φ δα = − 4 , H′ = H + 4 , I β β I β β ∂xα ∂ΠI ∂πI expl

which are in agreement with those of Eqs. (25).

3.6. Poisson brackets, Lagrange brackets

For a system with given Hamiltonian density H(φI ,ππI , x), and for two differentiable µ functions f(φI ,ππI , x), g(φI ,ππI , x) of the fields φI , πI and the independent variables xµ, we define the µ-th component of the Poisson bracket of f and g as follows ∂f ∂g ∂f ∂g f,g φφ,ππµ = µ − µ . (30) ∂φI ∂πI ∂πI ∂φI With this definition, the four Poisson brackets [f,g]φφ,ππµ constitute the components of a dual 4-vector, i.e., a 1-form. Obviously, the Poisson bracket (30) satisfies the following algebraic rules

f,g φφ,ππµ = − g,f φφ,ππµ R cf,g φφ,ππµ = c f,g φφ,ππµ , c ∈

f,g φφ,ππµ + h,gφφ,ππµ = f + h,g φφ,ππµ December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

16 J. Struckmeier, A. Redelbach

The Leibnitz rule is obtained from (30) via

∂f ∂ ∂f ∂ f,gh φφ,ππµ = µ (gh) − µ (gh) ∂φI ∂πI ∂πI ∂φI ∂f ∂g ∂h ∂f ∂h ∂g = h + g − g + h ∂φ ∂πµ ∂πµ ∂πµ ∂φ ∂φ I I I I I I ∂f ∂g ∂f ∂g ∂f ∂h ∂f ∂h = − h + g − ∂φ ∂πµ ∂πµ ∂φ ∂φ ∂πµ ∂πµ ∂φ I I I I I I I I = f,g φφ,ππµ h + g f,h φφ,ππµ For an arbitrary differentiable function f(φI ,ππI , x) of the field variables, we can, in particular, set up the Poisson brackets with the canonical fields φI , and πI . As the µ individual field variables φI and πI are independent by assumption, we immediately get

∂φI ∂f ∂φI ∂f ∂f ∂f φI ,f φφ,ππµ = µ − µ = δIJ µ = µ ∂φJ ∂πJ ∂πJ ∂φJ ∂πJ ∂πI ν ν ν ∂πI ∂f ∂πI ∂f ν ∂f ν ∂f πI ,f φφ,ππµ = µ − µ = −δIJ δµ = −δµ . ∂φJ ∂πJ ∂πJ ∂φJ ∂φJ ∂φI The Poisson bracket of a function f of the field variables with a particular field variable thus corresponds to the derivative of that function f with respect to the conjugate field variable. For the particular case f ≡ H, this means

∂H ∂φI φI , H φφ,ππµ = µ = µ ∂πI ∂x α ν ν ∂H ν ∂πI πI , H φφ,ππµ = −δµ = δµ α . (31) ∂φI ∂x The last equation reﬂects the fact that the covariant Hamiltonian density H only ν determines the divergence of the momentum vector πI and not the individual derivatives its components. This gives rise to the gauge freedom of the momentum vector ν πI , as discussed previously in Sect. 2.3,

β ∂πν ∂πν ∂πα ∂k πν , H = I − k ν , k ν = I − δν I ⇒ Iµ ≡ 0. I φφ,ππµ ∂xµ Iµ Iµ ∂xµ µ ∂xα ∂xβ

ν As the momentum vector πI is only determined by H up to divergence-free vectors ν ν ηI , the Poisson bracket [πI , H]φφ,ππµ is only determined up to divergence-free tensors ν ν kIµ . Thus, we can always choose an equivalent vectorπ ˜I for which

α ν ∂H ∂π˜I ν ∂π˜I − = α , π˜I , H φφ,π¯µ = µ . (32) ∂φI ∂x ∂x The fundamental Poisson brackets are constituted by pairing ﬁeld variables φI December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 17

µ and πI ,

∂φI ∂φJ ∂φI ∂φJ φI , φJ φφ,ππµ = µ − µ =0, ∂φK ∂πK ∂πK ∂φK ν ν ν ν ∂φI ∂πJ ∂φI ∂πJ ∂πJ ν φI , πJ φφ,ππµ = µ − µ = µ = δµ δIJ , (33) ∂φK ∂πK ∂πK ∂φK ∂πI α β α β α β ∂πI ∂πJ ∂πI ∂πJ πI , πJ φφ,ππµ = µ − µ =0. ∂φK ∂πK ∂πK ∂φK

Similar to point mechanics, we can define the Lagrange brackets as the dual coun- terparts of the Poisson brackets. In local description, we define the components of µ a 4-vector of Lagrange brackets {f,g}φφ,ππ of two differentiable functions f,g by

µ µ µ ∂φ ∂π ∂π ∂φ {f,g}φφ,ππ = I I − I I . (34) ∂f ∂g ∂f ∂g

The fundamental Lagrange bracket then emerge as

µ µ µ φφ,ππ ∂φK ∂πK ∂πK ∂φK φI , φJ = − =0, ∂φI ∂φJ ∂φI ∂φJ µ µ µ µ ν φφ,ππ ∂φK ∂πK ∂πK ∂φK ∂πI µ φI , πJ = ν − ν = ν = δν δIJ , (35) ∂φI ∂πJ ∂φI ∂πJ ∂πJ µ µ φφ,ππµ ∂φ ∂π ∂π ∂φ πα, πβ = K K − K K =0. I J ∂πα β ∂πα β I ∂πJ I ∂πJ

In the next section, we shall prove that both the Poisson brackets as well as the Lagrange brackets are invariant under canonical transformations of the ﬁelds φI ,ππI .

3.7. Canonical invariance of Poisson and Lagrange brackets In the ﬁrst instance, we will show that the fundamental Poisson brackets are invariant under canonical transformations, hence that the relations (33) equally apply for canonically transformed ﬁelds ΦI and ΠI . Making use of the symmetry rela- December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

18 J. Struckmeier, A. Redelbach

tions (20) and (26), one ﬁnds

∂ΦI ∂ΦJ ∂ΦI ∂ΦJ ΦI , ΦJ φφ,ππµ = µ − µ ∂φK ∂πK ∂πK ∂φK ∂ΦI ∂ΦJ ∂ΦI ∂ΦJ ν = µ − ν δµ ∂φK ∂πK ∂πK ∂φK ν ∂ΦI ∂φK ∂ΦI ∂πK = − µ − ν µ ∂φK ∂ΠJ ∂πK ∂ΠJ ∂ΦI = − µ =0= φI , φJ φφ,ππµ (36) ∂ΠJ ν ν ν ∂ΦI ∂ΠJ ∂ΦI ∂ΠJ ΦI , ΠJ φφ,ππµ = µ − µ ∂φK ∂πK ∂πK ∂φK ν ν ∂ΦI α ∂ΠJ ∂ΦI ∂ΠJ = δµ α − µ ∂φK ∂πK ∂πK ∂φK α ν ν ∂πK ∂ΠJ ∂φK ∂ΠJ = µ α + µ ∂ΠI ∂πK ∂ΠI ∂φK ν ∂ΠJ ν ν = µ = δµδIJ = φI , πJ φφ,ππµ (37) ∂ΠI α β In contrast, the Poisson bracket of a pair of momentum ﬁelds ΠI , ΠJ is not generally maintained. The symmetry relations (17) and (23) yield

α β α β α β ∂ΠI ∂ΠJ ∂ΠI ∂ΠJ ΠI , ΠJ φφ,ππµ = µ − µ ∂φK ∂πK ∂πK ∂φK α α β ∂ΠI ∂φK β ∂ΠI ∂πK = δµ + µ ∂φK ∂ΦJ ∂πK ∂ΦJ ∂Πα ∂Πα ∂πβ ∂πξ = I δβ + I K δξ − K δβ ∂Φ µ ξ ∂Φ µ ∂Φ µ J ∂πK J J =0 ∂Πα ∂Πξ ∂Πβ = | {zI} J δβ − J δξ ξ ∂φ µ ∂φ µ ∂πK K K 6= 0 in general. It vanishes for the particular subgroups of canonical transformations with transformed momentum fields ΠI not depending on either the original fields φK or on the original momentum fields πK , hence α α α β ∂ΠJ ∂ΠI ΠI , ΠJ φφ,ππµ = 0 if =0 or µ = 0 (38) ∂φK ∂πK for all indices α, µ = 0,..., 3. It can be concluded that the complete set of fundamental Poisson brackets is invariant if both

• the transformed ﬁelds ΦI do not depend on the original momentum ﬁelds πK according to the “point transformation condition” of Eq. (27) December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 19

• the transformed momentum ﬁelds ΠI do not depend on the original ﬁelds φK .

The configuration space — in which the fields φI , ΦI reside — and the “multi- cotangent” space of the momentum fields πI ,ΠI thus do not intermix under canonical transformations that are supposed to maintain the Poisson brackets. This requirement severely limits the set of valid transformations of fields and their conjugate momentum fields in the realm of canonical field theory. The Poisson bracket of two arbitrary differentiable functions f(φφ,πππ, x) and g(φφ,πππ, x), as defined by Eq. (30), can be expanded in terms of transformed fields ΦI and ΠI . For a general transformation (φφ,ππ) 7→ (ΦΦ,ΠΠ), we have ∂f ∂g ∂f ∂g f,g φφ,ππµ = µ − µ ∂φK ∂πK ∂πK ∂φK β ∂f ∂Φ ∂f ∂Πα ∂g ∂Φ ∂g ∂Π = I + I J + J ∂Φ ∂φ ∂Πα ∂φ ∂Φ ∂πµ β ∂πµ I K I K J K ∂ΠJ K ! ∂f ∂Φ ∂f ∂Πα ∂g ∂Φ ∂g ∂Πβ − I + I J + J . ∂Φ ∂πµ ∂Πα ∂πµ ∂Φ ∂φ β ∂φ I K I K J K ∂ΠJ K ! After working out the multiplications, we can recollect all products in terms of fundamental Poisson brackets ∂f ∂g ∂f ∂g f,g = Φ , Φ + Πα, Πβ φφ,ππµ ∂Φ ∂Φ I J φφ,ππµ ∂Πα β I J φφ,ππµ I J I ∂ΠJ ∂f ∂g ∂f ∂g + − Φ , Πα . ∂Φ ∂Πα ∂Πα ∂Φ I J φφ,ππµ I J J I For the special case that the transformation is canonical , the equations (36), (37), and (38) for the fundamental Poisson brackets apply. We then get ∂f ∂g ∂f ∂g f,g = − δαδ = f,g . (39) φφ,ππµ ∂Φ ∂Πα ∂Πα ∂Φ µ IJ Φ,Πµ I J J I We thus abbreviate in the following the index notation of the Poisson bracket by

writing f,g µ ≡ f,g φφ,ππµ , as the brackets do not depend on the underlying set of canonical ﬁeld variables φ ,ππ . I I The proof of the canonical invariance of the ﬁrst and second fundamental La- grange bracket is based on the symmetry relations (17) and (23), namely

µ µ µ φφ,ππ ∂φK ∂πK ∂πK ∂φK ΦI , ΦJ = − ∂ΦI ∂ΦJ ∂ΦI ∂ΦJ µ ν ∂φK ∂ΠJ ∂πK ∂φK µ = − − δν ∂ΦI ∂φK ∂ΦI ∂ΦJ µ ν µ ∂φK ∂ΠJ ∂πK ∂ΠJ = − − ν ∂ΦI ∂φK ∂ΦI ∂πK µ µ ∂ΠJ φφ,ππ = − =0= φI , φJ (40) ∂ΦI December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

20 J. Struckmeier, A. Redelbach

µ µ µ ν φφ,ππ ∂φK ∂πK ∂πK ∂φK ΦI , ΠJ = ν − ν ∂ΦI ∂ΠJ ∂ΦI ∂ΠJ α µ ∂φK µ ∂πK ∂ΠI ∂φK = δα ν + ν ∂ΦI ∂ΠJ ∂φK ∂ΠJ µ α µ ∂ΠI ∂πK ∂ΠI ∂φK = α ν + ν ∂πK ∂ΠJ ∂φK ∂ΠJ µ µ ∂ΠI µ ν φφ,ππ = ν = δν δIJ = φI , πJ (41) ∂ΠJ In contrast to the Poisson bracket, the Lagrange bracket of a pair of momentum α β ﬁelds ΠI , ΠJ is generally maintained. The symmetry relations (20) and (26) yield

µ µ φφ,ππµ ∂φ ∂π ∂π ∂φ Πα, Πβ = K K − K K I J ∂Πα β ∂Πα β I ∂ΠJ I ∂ΠJ ∂φ ∂Φ ∂πξ ∂φ = K J δµ − K K δµ ∂Πα ∂φ β ∂Πα β ξ I K I ∂ΠJ ∂φ ∂Φ ∂πξ ∂Φ = K J δµ + K J δµ ∂Πα ∂φ β ∂Πα ξ β I K I ∂πK ∂ΦJ µ = α δβ ∂ΠI =0. (42)

The Lagrange bracket (34) of two arbitrary diﬀerentiable functions f(φφ,πππ, x) and g(φφ,πππ, x) can be expressed in terms of transformed ﬁelds ΦI , ΠI

µ µ φφ,ππµ ∂φ ∂π ∂π ∂φ f,g = K K − K K ∂f ∂g ∂f ∂g ∂φ ∂Φ ∂φ ∂Πα ∂πµ ∂Φ ∂πµ ∂Πβ = K I + K I K J + K J ∂Φ ∂f ∂Πα ∂f ∂Φ ∂g β ∂g I I J ∂ΠJ ! ∂πµ ∂Φ ∂πµ ∂Πα ∂φ ∂Φ ∂φ ∂Πβ − K I + K I K J + K J . ∂Φ ∂f ∂Πα ∂f ∂Φ ∂g β ∂g I I J ∂ΠJ ! Multiplication and regathering the terms to form fundamental Lagrange brackets yields

α β φφ,ππµ ∂Φ ∂Φ φφ,ππµ ∂Π ∂Π φφ,ππµ f,g = I J Φ , Φ + I J Πα, Πβ ∂f ∂g I J ∂f ∂g I J β α ∂Φ ∂Π φφ,ππµ ∂Π ∂Φ φφ,ππµ + I J Φ , Πβ − I J Φ , Πα . ∂f ∂g I J ∂f ∂g J I For canonical transformations, we can make use of the relations (40), (41), and (42) December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 21

for the fundamental Lagrange brackets. We thus obtain β α φφ,ππµ ∂Φ ∂Π ∂Π ∂Φ f,g = I J δ δµ − I J δ δµ ∂f ∂g IJ β ∂f ∂g IJ α ∂Φ ∂Πµ ∂Πµ ∂Φ = I I − I I ∂f ∂g ∂f ∂g Φ,Πµ = f,g . The notation of the Lagrange brackets (34) can thus be simpliﬁed as well. In the µ following, we denote these brackets as f,g since their value does not depend on the particular set of canonical ﬁeld variables φ ,ππ . I I

3.8. Liouville’s theorem of covariant Hamiltonian field theory Based on the theory of canonical transformations from Sect. 3, we may express Li- ouville’s theorem of covariant field theory in the following general way: the volume µ µ form dV = dφ1 . . . dφN dπ1 . . . dπI of a Hamiltonian system of N fields φI is invariant with respect to canonical transformations for each individual index µ =0,..., 3 of the set of independent variables

µ µ can. transf. µ µ dφ1 . . . dφN dπ1 . . . dπN = dΦ1 ...dΦN dπ1 . . . dπN . For general transformations of the system’s coordinates, the transformation of the volume form dV is determined by the determinant det J of the associated Jacobi matrix J µ µ µ µ dφ1 . . . dφN dπ1 . . . dπN = det J dΦ1 ...dΦN dπ1 . . . dπN µ µ ∂(φ1,...,φN , π1 ,...,πN ) det J = µ µ . ∂(Φ1,..., ΦN , π1 ,...,πN ) Liouville’s theorem thus states that the determinant det J of the transformation’s Jacobi matrix equals one, | det J| = 1, in case that the transformation is canonical. To prove this, we write det J in explicit form,

∂φ1 ∂φ1 ∂φ1 ∂φ1 ... µ ... µ ∂Φ1 ∂ΦN ∂π1 ∂πN ...... ∂φN ∂φN ∂φN ∂φN ... µ ... µ ∂Φ1 ∂ΦN ∂π1 ∂πN det J = µ µ µ µ , ∂π1 ∂π1 ∂π1 ∂π1 ∂Φ ... ∂Φ ∂Πµ ... ∂Πµ 1 N 1 N ...... µ µ µ µ ∂πN ∂πN ∂πN ∂πN ∂Φ ... ∂Φ ∂πµ ... ∂πµ 1 N 1 N

where no summation over µ is understood. In terms of a 2 × 2 block matrix, det J

can be written concisely as the determinant A B det J = (43) CD

December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

22 J. Struckmeier, A. Redelbach

with the four N × N blocks A, B, C, and D deﬁned by ∂φ ∂φ A = I , B = I ∂Φ ∂πµ K K ∂πµ ∂πµ C = I ,D = I . (44) ∂Φ ∂πµ K K Herein, K = 1,...,N denotes the column index, and I = 1,...,N the row index, respectively, whereas µ ∈ [0,..., 3] is held ﬁxed. For the inverse transformation, we pursue a similar procedure

µ µ −1 µ µ dΦ1 ...dΦN dπ1 . . . dπN = (det J) dφ1 . . . dφN dπ1 . . . dπN µ µ −1 ∂(Φ1,..., ΦN , π1 ,...,πN ) (det J) = µ µ . ∂(φ1,...,φN , π1 ,...,πN )

The explicit form of (det J)−1

∂Φ1 ∂Φ1 ∂Φ1 ∂Φ1 ... µ ... µ ∂φ1 ∂φN ∂π1 ∂πN ...... ∂ΦN ∂ΦN ∂ΦN ∂ΦN ∂φ ... ∂φ ∂πµ ... ∂πµ −1 1 N 1 N (det J) = µ µ µ µ ∂π1 ∂π1 ∂π1 ∂π1 ∂φ ... ∂φ ∂πµ ... ∂πµ 1 N 1 N ...... µ µ µ µ ∂πN ∂πN ∂πN ∂πN ∂φ ... ∂φ ∂πµ ... ∂πµ 1 N 1 N

can be written equivalently as the determinant of a 2 × 2 block matrix E F (det J)−1 = (45) G H

with the four N × N blocks E, F , G, and H (no summation over µ!) given by ∂Φ ∂Φ E = I , F = I ∂φ ∂πµ K K ∂πµ ∂πµ G = I , H = I . (46) ∂φ ∂πµ K K Making use of the symmetry relations (17), (20) (23), and (26) that apply exactly if the respective transformation is canonical, the following identities hold for the block matrices (44) and (46)

∂πµ ∂φ E = K = DT , F = − K = −BT ∂πµ ∂πµ I I ∂πµ ∂φ G = − K = −CT , H = K = AT . (47) ∂Φ ∂Φ I I December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 23

We shall now show that det J and (det J)−1 must be equal. The determinant of the block matrix J can be expressed in terms of the determinants of its blocks as

det J = det A det S, S = D − CA−1B. (48) (det J)−1 = det H det S,˜ S˜ = E − FH−1G. (49)

The matrix S is referred to as the “Schur complement” of A, and, correspondingly, S˜ as the Schur complement of H. Starting from formula (49) we ﬁnd, inserting the identities (47),

(det J)−1 = det H det E − FH−1G −1 = det AT det DT − BT AT CT

h T i = det AT det DT − BT A−1 CT h T i = det AT det D − CA−1B −1 = det A det D − CA B = det J

according to Eq. (48). From (det J)−1 = det J, we ﬁnally conclude that

| det J| =1, (50)

which proves the assertion.

3.9. Jacobi’s identity and Poisson’s theorem in canonical field theory In order to derive the canonical field theory analog of Jacobi’s identity of point mechanics, we let f(φφ,πππ, x), g(φφ,πππ, x), and h(φφ,πππ, x) denote arbitrary differentiable functions of the canonical fields. The sum of the three cyclicly permuted nested Poisson brackets be denoted by aµν ,

aµν = f, g,h + h, f,g + g, h,f . (51) µ ν µ ν µ ν h i h i h i We will now show that the aµν are the components of a skew-symmetric (0, 2) tensor, hence that

aµν + aνµ =0. (52)

Writing Eq. (51) explicitly, we get a sum of 24 terms, each of them consisting of a triple product of two ﬁrst-order derivatives and one second-order derivative of the December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

24 J. Struckmeier, A. Redelbach

functions f, g, and h

∂f ∂ ∂g ∂h ∂g ∂h a = − µν ∂φ ∂πν ∂φ ∂πµ ∂πµ ∂φ J J I I I I ∂f ∂ ∂g ∂h ∂g ∂h − − ∂πν ∂φ ∂φ ∂πµ ∂πµ ∂φ J J I I I I ∂h ∂ ∂f ∂g ∂f ∂g + − ∂φ ∂πν ∂φ ∂πµ ∂πµ ∂φ J J I I I I ∂h ∂ ∂f ∂g ∂f ∂g − − ∂πν ∂φ ∂φ ∂πµ ∂πµ ∂φ J J I I I I ∂g ∂ ∂h ∂f ∂h ∂f + − ∂φ ∂πν ∂φ ∂πµ ∂πµ ∂φ J J I I I I ∂g ∂ ∂h ∂f ∂h ∂f − − . ∂πν ∂φ ∂φ ∂πµ ∂πµ ∂φ J J I I I I

The proof can be simplified making use of the fact that the terms of aµν from Eq. (51) emerge as cyclic permutations of the functions f, g, and h. With regard to the explicit form of Eq. (51) from above it suffices to show that Eq. (52) is fulfilled for all terms containing second derivatives of, for instance, f,

∂h ∂g ∂2f ∂h ∂g ∂2f aµν = µ ν − ν µ ∂φJ ∂πI ∂πJ ∂φI ∂φJ ∂φI ∂πJ ∂πI ∂h ∂g ∂2f ∂h ∂g ∂2f − ν µ + ν µ ∂πJ ∂πI ∂φJ ∂φI ∂πJ ∂φI ∂φJ ∂πI ∂g ∂h ∂2f ∂g ∂h ∂2f + ν µ − µ ν ∂φJ ∂φI ∂πJ ∂πI ∂φJ ∂πI ∂πJ ∂φI ∂g ∂h ∂2f ∂g ∂h ∂2f − ν µ + ν µ + ... (53) ∂πJ ∂φI ∂φJ ∂πI ∂πJ ∂πI ∂φJ ∂φI Resorting and interchanging the sequence of diﬀerentiations yields

∂h ∂g ∂2f ∂h ∂g ∂2f aµν = − ν µ + µ ν ∂φI ∂πJ ∂πI ∂φJ ∂φI ∂φJ ∂πI ∂πJ ∂h ∂g ∂2f ∂h ∂g ∂2f + µ ν − µ ν ∂πI ∂πJ ∂φI ∂φJ ∂πI ∂φJ ∂φI ∂πJ ∂g ∂h ∂2f ∂g ∂h ∂2f − µ ν + ν µ ∂φI ∂φJ ∂πI ∂πJ ∂φI ∂πJ ∂πI ∂φJ ∂g ∂h ∂2f ∂g ∂h ∂2f + µ ν − µ ν + ... (54) ∂πI ∂φJ ∂φI ∂πJ ∂πI ∂πJ ∂φI ∂φJ Mutually renaming the formal summation indices I and J, the right hand sides of Eqs. (53) and (54) diﬀer only by the sign and the interchange of the indices µ and ν. Thereby, Eq. (52) is proved. December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 25

Poisson’s theorem in the realm of canonical field theory is based on the identity ∂ ∂f ∂g f,g = ,g + f, . (55) ∂xν µ ∂xν µ ∂xν µ h i h i In contrast to point mechanics, this identity is most easily proved directly, i.e., without referring to the Jacobi identity (52). From to the definition (30) of the Poisson brackets, we conclude for two arbitrary differentiable functions f(φφ,πππ, x) and g(φφ,πππ, x) ∂ ∂ ∂f ∂g ∂f ∂g f,g = − ∂xν µ ∂xν ∂φ ∂πµ ∂πµ ∂φ I I I I ∂g ∂ ∂f ∂f ∂ ∂g = + ∂πµ ∂xν ∂φ ∂φ ∂xν ∂πµ I I I I ∂g ∂ ∂f ∂f ∂ ∂g − − ∂φ ∂xν ∂πµ ∂πµ ∂xν ∂φ I I I I ∂g ∂2f ∂φ ∂2f ∂πα ∂2f = J + J + ∂πµ ∂φ ∂φ ∂xν ∂φ ∂πα ∂xν ∂φ ∂xν I I J I J I expl!

∂f ∂2g ∂φ ∂2g ∂πα ∂2g + J + J + ∂φ ∂πµ∂φ ∂xν ∂πµ∂πα ∂xν ∂πµ∂xν I I J I J I expl!

∂g ∂2f ∂φ ∂2f ∂πα ∂2f − J + J + ∂φ ∂πµ∂φ ∂xν ∂πµ∂πα ∂xν ∂πµ∂xν I I J I J I expl!

∂f ∂2g ∂φ ∂2g ∂πα ∂2g − J + J + ∂πµ ∂φ ∂φ ∂xν ∂φ ∂πα ∂xν ∂φ ∂xν I I J I J I expl!

∂g ∂ ∂f ∂φ ∂f ∂πα ∂f = J + J + ∂πµ ∂φ ∂φ ∂xν ∂πα ∂xν ∂xν I I J J expl!

∂g ∂ ∂f ∂φ ∂f ∂πα ∂f − J + J + ∂φ ∂πµ ∂φ ∂xν ∂πα ∂xν ∂xν I I J J expl!

∂f ∂ ∂g ∂φ ∂g ∂πα ∂g + J + J + ∂φ ∂πµ ∂φ ∂xν ∂πα ∂xν ∂xν I I J J expl!

∂f ∂ ∂g ∂φ ∂g ∂πα ∂g − J + J + ∂πµ ∂φ ∂φ ∂xν ∂πα ∂xν ∂xν I I J J expl!

∂g ∂ ∂f ∂g ∂ ∂f = − ∂πµ ∂φ ∂xν ∂φ ∂πµ ∂xν I I I I ∂f ∂ ∂g ∂f ∂ ∂g + − ∂φ ∂πµ ∂xν ∂πµ ∂φ ∂xν I I I I ∂f ∂g = ,g + f, . ∂xν µ ∂xν µ h i h i Provided that both the ﬁrst derivative ∂/∂xν as well as the two second deriva- December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

26 J. Struckmeier, A. Redelbach

2 ν 2 µ ν tives ∂ /∂φI ∂x and ∂ /∂πI ∂x vanish for both functions f and g, then the ﬁrst ν derivative with respect to x of the Poisson bracket [f,g]µ also vanishes ∂f ∂g ∂2f ∂2f ∂ ν =0, ν =0, ν =0, µ ν =0 =⇒ ν f,g µ =0. ∂x ∂x ∂φI ∂x ∂πI ∂x ∂x This establishes Poisson’s theorem for canonical ﬁeld theory.

3.10. Hamilton-Jacobi equation In the realm of canonical ﬁeld theory, we can set up the Hamilton-Jacobi equation as µ follows: we look for a generating function F1 (φφ,Φ, x) of a canonical transformation that maps a given Hamiltonian density H into a transformed density that vanishes identically, H′ ≡ 0. In the transformed system, all partial derivatives of H′ thus vanish as well — and hence the derivatives of all ﬁelds ΦI (x), ΠI (x) with respect to the system’s independent variables xµ, ′ α ′ ∂H ∂ΠI ∂H ∂ΦI =0= α , µ =0= µ . ∂ΦI ∂x ∂ΠI ∂x According to the transformation rules (16) that arise from a generating function of type S ≡ F 1(φφ,Φ, x), this means for a given Hamiltonian density H ∂Sα H(φφ,πππ, x)+ =0. (56) ∂xα expl

µ µ In conjunction with the transformation rule πI = ∂S /∂φI , we may subsequently set up the Hamilton-Jacobi equation as a partial diﬀerential equation for the 4- vector function S ∂SS ∂Sα H φφ, , x + =0. ∂φφ ∂xα expl

This equation illustrates that the generating function S defines exactly that particular canonical transformation which maps the space-time state of the system into its fixed initial state µ µ ΦI = φI (000) = const., ΠI = πI (000) = const. The inverse transformation then defines the mapping of the system’s initial state into its actual state in space-time. ′ µ As a result of the fact that H as well as all ∂ΦI /∂x vanish, the divergence of S(φφ,Φ, x) simplifies to ∂Sα ∂Sα ∂φ ∂Sα = I + ∂xα ∂φ ∂xα ∂xα I expl

α ∂φI = π − H I ∂xα = L. December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 27

This equation coincides with the transformation rule (14) of the Lagrangians for the particular case L′ ≡ 0. The 4-vector function S thus embodies the ﬁeld theory analogue of Hamilton’s principal function S, dS/dt = L of point mechanics.

4. Examples for Hamiltonian densities in covariant ﬁeld theory 4.1. Ginzburg-Landau Hamiltonian density We consider the scalar ﬁeld φ(x, t) whose Lagrangian density L is given by

1 2 2 2 2 2 L(φ, ∂tφ, ∂xφ)= 2 (∂tφ) − v (∂xφ) + λ φ − 1 . (57) Herein, v and λ are supposed toh denote constant quantities.i The particular Euler- Lagrange equation for this Lagrangian density simplifies the general form of Eq. (2) to ∂ ∂L ∂ ∂L ∂L + − =0. ∂t ∂(∂tφ) ∂x ∂(∂xφ) ∂φ The resulting field equation is ∂2φ ∂2φ − v2 − 4λ φ φ2 − 1 =0. (58) ∂t2 ∂x2 In order to derive the equivalent Hamiltonian representation, we first define the conjugate momentum fields from L

∂L ∂φ ∂L 2 ∂φ πt(x, t)= = , πx(x, t)= = −v . ∂(∂tφ) ∂t ∂(∂xφ) ∂x The Hamiltonian density H now follows as the Legendre transform of the La- grangian density L ∂φ ∂φ H(φ, π , π )= π + π − L(φ, ∂ φ, ∂ φ). t x t ∂t x ∂x t x The Ginzburg-Landau Hamiltonian density H is thus given by

1 2 H(φ, π , π )= 1 π2 − π2 − λ φ2 − 1 . (59) t x 2 t v2 x The canonical field equations for the density H of Eq. (59) are ∂H ∂φ ∂H ∂φ ∂H ∂π ∂π = , = , = − t − x , ∂πt ∂t ∂πx ∂x ∂φ ∂t ∂x from which we derive the following set coupled first order equations ∂φ ∂φ ∂π ∂π π = , π = −v2 , t + x − 4λ φ φ2 − 1 =0. t ∂t x ∂x ∂t ∂x As usual, the canonical field equations for the scalar field φ(x, t) just reproduce the definition of the momentum fields πt and πx from the Lagrangian density L. By inserting πt and πx into the second field equation the coupled set of first order field equations is converted into a single second order equation for φ(x, t): ∂2φ ∂2φ − v2 − 4λ φ φ2 − 1 =0, ∂t2 ∂x2 which coincides with Eq. (58), as expected. December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

28 J. Struckmeier, A. Redelbach

4.2. Klein-Gordon Hamiltonian density for a real scalar ﬁeld

We ﬁrst consider the Klein-Gordon Lagrangian density LKG for a real scalar ﬁeld φ(x)

1 ∂φ ∂φ 1 2 2 LKG (φ, ∂µφ)= 2 α − 2 ω φ (x). (60) ∂x ∂xα The Euler-Lagrange equation (2) for φ(x) follows from this Lagrangian density as 2 ∂ φ 2 α = −ω φ. (61) ∂xα∂x

We now derive the corresponding covariant Hamiltonian density HKG(φ, πµ) that contains the identical information on the dynamical system and thus yields with the covariant canonical equations (5) the equivalent description of the system’s dynamics. To this end, we must first define from LKG the conjugate momentum fields,

µ ∂LKG ∂φ ∂LKG ∂φ π = = , πµ = = µ . ∂φ ∂xµ ∂φ ∂x ∂ µ ∂ ∂x ∂xµ The Hamiltonian density Hthen follows as the Legendre transform of the La- grangian density ∂φ H(φ, πµ)= πα − L(φ, ∂ φ), ∂xα µ provided that the Hesse matrix that is associated with any Legendre transformation is non-singular. For the actual case, the components of the Hesse matrix are ∂2L KG = δµ, ∂φ ∂φ ν ∂ µ ∂ ∂x ∂xν hence establish the unit matrix. The covariant Klein-Gordon Hamiltonian density HKG thus exists and is given by µ 1 α 1 2 2 HKG(φ, π )= 2 παπ + 2 ω φ (x). (62) For the Hamiltonian density (62), the canonical field equations (5) provide the following set of coupled first order partial differential equations α ∂φ ∂HKG µ ∂π ∂HKG 2 = = π , − α = = ω φ. (63) ∂xµ ∂πµ ∂x ∂φ Obviously, the canonical field equation for the scalar field φ(x) coincides with the definition of the momentum field πµ(x) from the Lagrangian density LKG. Inserting πµ from the first canonical field equation into the second then yields the Euler- Lagrange equation of Eq. (61). We note that the Hamiltonian density HKG resembles the Hamiltonian function H of a conservative Hamiltonian system of classical particle mechanics, which is given by H = T + V as the sum of kinetic energy T and potential energy V . In this December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 29

regard, the Klein-Gordon equation is nothing else as the ﬁeld theory analog of the equation of motion of the harmonic oscillator of point mechanics. The Heisenberg equations (31) compute to ∂φ = [φ, H ] = φ, 1 παπ + 1 ω2 φ2 ∂xµ KG µ 2 α 2 µ α 2 = [φ, π ]µ πα +ω [φ, φ]µ φ

α =δµ =0

= |πµ {z } | {z }

and ∂πα δν = [πν , H ] = πν , 1 παπ + 1 ω2 φ2 µ ∂xα KG µ 2 α 2 µ ν α 2 ν = [π , π ]µ πα +ω [π , φ]µ φ

ν =0 =−δµ ν 2 = |−δµ{zω φ} | {z } α 2 ∂π 2 ∂ φ 2 ⇒ α = −ω φ, α + ω φ =0. ∂x ∂xα∂x As expected, the Klein-Gordon ﬁeld equation emerges.

4.3. Klein-Gordon Hamiltonian density for a set of N complex scalar ﬁelds

We ﬁrst consider the Klein-Gordon Lagrangian density LKG for a set of complex 1 scalar ﬁelds φI (see, for instance, Ref. ):

µ ∂φI ∂φI 2 LKG φφ,φφ, ∂ φφ, ∂µφ = α − φI ωIJ φJ . (64) ∂x ∂xα 2 Herein φI denotes the complex conjugate ﬁeld of φI and ωIJ the (diagonal) mass matrix. The quantities φI and φI are to be treated as independent. The Euler- Lagrange equations (2) for φI and φI follow from this Lagrangian density as

2 2 ∂ φI 2 ∂ φI 2 α = −φJ ωJI , α = −ωIJ φJ . (65) ∂xα∂x ∂xα∂x

As a prerequisite for deriving the corresponding Hamiltonian density HKG we must first define from LKG the conjugate momentum fields,

µ ∂LKG ∂φI ∂LKG ∂φI πI = = , πIµ = = µ . ∂φI ∂xµ ∂φI ∂x ∂ µ ∂ ∂x ∂xµ

µ Thereby, the canonical momenta πI are deﬁned as the dual quantities of the partial derivatives of the ﬁelds φI . December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

30 J. Struckmeier, A. Redelbach

The Hamiltonian density H then follows as the Legendre transform of the La- grangian density

∂φ ∂φ H(π µ,πµ,φφφ,φ)= I πα + πα I − L. ∂xα I I ∂xα

The Klein-Gordon Hamiltonian density HKG is thus given by

µ α 2 HKG(π µ,π ,φφφ,φ)= πIαπI + φI ωIJ φJ . (66)

For the Hamiltonian density (66), the canonical field equations (5) provide the following set of coupled first order partial differential equations

∂HKG ∂φI ∂HKG ∂φI µ µ = µ = πIµ, = = πI ∂πI ∂x ∂πIµ ∂xµ α α ∂HKG ∂πI 2 ∂HKG ∂πI 2 = − α = φJ ωJI , = − α = ωIJ φJ . (67) ∂φI ∂x ∂φI ∂x As a result of the Legendre transformation, the canonical ﬁeld equations for the

scalar fields φI and φI coincide with the definitions of the momentum fields πIµ and µ µ πI from the Lagrangian density LKG. Eliminating the πIµ, πI from the canonical field equations then yields the Euler-Lagrange equations of Eq. (65). For complex fields, the energy-momentum tensor in the Lagrangian formalism is defined analogously to the real case of Eq. (6)

ν ∂L ∂φI ∂φI ∂L ν θµ = + µ − δµ L. ∂φI ∂xν ∂x ∂φI ∂ ∂ ν ∂xµ ∂x Expressed by means of the complex Hamiltonian density H, this means

ν ∂φI ∂φI ν ν α ∂φI ν ∂φI α ν θµ = πIµ + µ πI − δµ πI α − δµ α πI + δµ H. ∂xν ∂x ∂x ∂x

For the Klein-Gordon Hamiltonian density HKG from Eq. (66), we thus get the ν particular canonical energy-momentum tensor θµ

ν ν ν α ν 2 θµ =2πI πIµ − δµ πI πIα + δµ φI ωIJ φJ . (68)

The equations of motion for the sets of ﬁelds φI ,ππI can be derived as well from the covariant Heisenberg equations (31)

∂φ ∂πα ∂φ ∂πα I = φ , H , δν I = πν , H , I = φ , H , δν I = πν , H , ∂xµ I µ µ ∂xα I µ ∂xµ I µ µ ∂xα I µ in conjunction with the fundamental Poisson bracket rules from Eqs. (33). Here,

the fields (φI ,ππI ) and, correspondingly, ( φI ,π I ) represent the pairs of canonical conjugate quantities, while φI and φI are to be treated as independent fields. This applies as well for the conjugate fields, πI and π I . The equations for φI and φI December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 31

follows as ∂φ I = [φ , H ] = φ , π πα + φ ω2 φ ∂xµ I KG µ I Jα J J JK K µ α α 2 2 = πJα [φI , πJ ]µ + [φI , πJ ]µ πJα + φJ ωJK [φI , φK ]µ + φI , φJ µ ωJK φK

α =0 =δµ δIJ =0 =0

= πIµ.| {z } | {z } | {z } | {z } (69)

∂φ I = φ , H = φ , π πα + φ ω2 φ ∂xµ I KG µ I Jα J J JK K µ α α 2 2 = πJα φI , πJ µ + φI , πJ µ πJα + φJ ωJK φI , φK µ + φI , φJ µ ωJK φK

α =δµ δIJ =0 =0 =0

= πIµ.| {z } | {z } | {z } | {z } (70)

Similarly, the equations for the divergences of πI and πI are ∂πα δν I = [πν , H ] = πν , παπ + φ ω2 φ µ ∂xα I KG µ I J Jα J JK K µ ν α ν α 2 ν ν 2 = πJα [πI , πJ ]µ + [πI , πJ ]µ πJα + φJ ωJK [πI , φK ]µ + πI , φJ µ ωJK φK

=0 =0 =0 ν =−δµδIJ ν 2 = −δµ |ωIK{zφK} | {z } | {z } | {z } (71)

∂πα δν I = [πν , H ] = πν , παπ + φ ω2 φ µ ∂xα I KG µ I J Jα J JK K µ ν α ν α 2 ν ν 2 = πJα [πI , πJ ]µ + [πI , πJ ]µ πJα + φJ ωJK [πI , φK ]µ + πI , φJ µ ωJK φK

ν =0 =0 =−δµδIK =0 ν 2 = −δµ |φJ ω{zJI .} | {z } | {z } | {z } (72) Obviously, the obtained equations coincide with the canonical equations from Eqs. (67).

4.4. Maxwell’s equations as canonical ﬁeld equations

The Lagrangian density LM of the electromagnetic field is given by 4π ∂a ∂a L (aa,∂∂∂aaa, x)= − 1 f f αβ − jα(x) a , f = ν − µ . (73) M 4 αβ c α µν ∂xµ ∂xν Herein, the four components aµ of the 4-vector potential a now take the place of the µ scalar fields φI ≡ a in the notation used so far. The Lagrangian density (73) thus entails a set of four Euler-Lagrange equations, i.e., an equation for each component aµ. The source vector j = (cρ,jx, jy, jz) denotes the 4-vector of electric currents combining the usual current density vector (jx, jy, jz) of configuration space with December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

32 J. Struckmeier, A. Redelbach

the charge density ρ. In a local Lorentz frame, i.e., in Minkowski space, the Euler- Lagrange equations (2) take on the form,

∂ ∂LM ∂LM α − =0, µ =0,..., 3. (74) ∂x ∂(∂αaµ) ∂aµ

With LM from Eq. (73), we obtain directly ∂f µα 4π + jµ =0. (75) ∂xα c In Minkowski space, this is the tensor form of the inhomogeneous Maxwell equation. In order to formulate the equivalent Hamiltonian description, we first define, according to Eq. (3), the canonically field components pµν as the conjugate objects of the derivatives of the 4-vector potential a ∂L ∂L pµν = M ≡ M (76) ∂(∂ν aµ) ∂aµ,ν With the particular Lagrangian density (73), Eq. (76) means, in detail,

∂f ∂f αβ pµν = − 1 αβ f αβ + f 4 ∂(∂ a ) ∂(∂ a ) αβ ν µ ν µ ∂f ∂f = − 1 αβ f αβ + αβ f αβ 4 ∂(∂ a ) ∂(∂ a ) ν µ ν µ 1 αβ ∂fαβ = − 2 f ∂(∂ν aµ)

1 αβ ∂ ∂aβ ∂aα = − 2 f α − β ∂aµ ∂x ∂x ∂ ∂xν

1 αβ µ ν µ ν 1 νµ µν = − 2 f δβ δα − δαδβ = − 2 (f − f ) = f µν .

The tensor pµν thus matches exactly the electromagnetic field tensor f µν from Eq. (73) and hence inherits the skew-symmetry of f µν because of the particular ν dependence of LM on the aµ,ν ≡ ∂aµ/∂x . As the Lagrangian density (73) now describes the dynamics of a vector field, µν aµ, rather than a set of scalar fields φI , the canonical momenta p now constitute ν a second rank tensor rather than a vector, πI The Legendre transformation cor- αβ responding to Eq. (4) then comprises the product p ∂βaα. The skew-symmetry µν of the momentum tensor p picks out the skew-symmetric part of ∂ν aµ as the αβ symmetric part of ∂ν aµ vanishes identically calculating the product p ∂βaα

∂a ∂a ∂a ∂a ∂a pαβ α = 1 pαβ α − β + 1 pαβ α + β . ∂xβ 2 ∂xβ ∂xα 2 ∂xβ ∂xα ≡0 | {z } December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 33

For a skew-symmetric momentum tensor pµν , we thus obtain the Hamiltonian density HM as the Legendre-transformed Lagrangian density LM

∂a ∂a H (aa,ppp, x)= 1 pαβ α − β − L (aa,∂∂∂aaa, x). M 2 ∂xβ ∂xα M From this Legendre transformation prescription and the Euler-Lagrange equations (74), the canonical ﬁeld equations are immediately obtained as

∂H 1 ∂a ∂a M = µ − ν ∂pµν 2 ∂xν ∂xµ µα ∂HM ∂LM ∂ ∂LM ∂p = − = − α = − α ∂aµ ∂aµ ∂x ∂(∂αaµ) ∂x ∂H ∂L M = − M . ∂xν ∂xν The Hamiltonian density for the Lagrangian density (73) follows as

4π H (aa,ppp, x)= − 1 pαβp + 1 pαβp + jα(x) a M 2 αβ 4 αβ c α 4π = − 1 pαβp + jα(x) a , pαβ = −pβα. (77) 4 αβ c α The ﬁrst canonical ﬁeld equation follows from the derivative of the Hamiltonian density (77) with respect to pµν

∂a ∂H µ = M = − 1 p = 1 p , ∂xν ∂pµν 2 µν 2 νµ

hence ∂a ∂a p = ν − µ , (78) µν ∂xµ ∂xν

µν which reproduces the deﬁnition of pµν and p from Eq. (76), as a consequence of the Legendre transformation. The second canonical ﬁeld equation is obtained calculating the derivative of the Hamiltonian density (77) with respect to aµ

µα ∂p ∂HM 4π µ − α = = j . ∂x ∂aµ c

Inserting the ﬁrst canonical equation, the second order ﬁeld equation for the aµ is thus obtained for the Maxwell Hamiltonian density (77) as

∂f µα 4π + jµ =0, (79) ∂xα c which agrees, as expected, with the corresponding Euler-Lagrange equation (75). December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

34 J. Struckmeier, A. Redelbach

4.5. The Proca Hamiltonian density In relativistic quantum theory, the dynamics of particles of spin 1 and mass m is derived from the Proca Lagrangian density LP, ∂a ∂a mc L = − 1 f αβf + 1 ω2aαa , f = ν − µ , ω = . (80) P 4 αβ 2 α µν ∂xµ ∂xν ~

We observe that the kinetic term of LP agrees with that of the Lagrangian density LM of the electromagnetic field of Eq. (73). Therefore, the field equations emerging from the Euler-Lagrange equations (74) are similar to those of Eq. (75) ∂f µα − ω2aµ =0. (81) ∂xα The transition to the corresponding Hamilton description is performed by defining µν on the basis of the actual Lagrangian LP the canonical momentum field tensors p as the conjugate objects of the derivatives of the 4-vector a ∂L ∂L pµν = P ≡ P . ∂(∂ν aµ) ∂aµ,ν Similar to the preceding section, we find

µν µν p = f , pµν = fµν

µ αβ because of the particular dependence of LP on the derivatives of the a . With p αβ being skew-symmetric in α, β, the product p aα,β picks out the skew-symmetric β part of the partial derivative ∂aα/∂x as the product with the symmetric part vanishes identically. Denoting the skew-symmetric part by a[α,β], the Legendre transformation prescription

αβ HP = p aα,β − LP αβ = p a[α,β] − LP ∂a ∂a = 1 pαβ α − β − L , 2 ∂xβ ∂xα P leads to the Proca Hamiltonian density by following the path of Eq. (77)

1 αβ 1 2 α αβ βα HP = − 4 p pαβ − 2 ω a aα, p = −p . (82) The canonical field equations emerge as ∂a ∂H µ = P = − 1 p = 1 p ∂xν ∂pµν 2 µν 2 νµ µα ∂p ∂HP 2 µ − α = = −ω a . ∂x ∂aµ By means of eliminating pµν , this coupled set of first order equations can be converted into second order equations for the vector field a(x), ∂ ∂a ∂a µ − α − ω2a =0. ∂x ∂xα ∂xµ µ α December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 35

As expected, this equation coincides with the Euler-Lagrange equation (81). νµ To derive the equations of motion for the ﬁelds aν ,p using the covariant Heisenberg equations (31) ∂a ∂pνα ν = a , H , δξ = pνξ, H , ∂xµ ν µ µ ∂xα µ we must apply the fundamental Poisson bracket rules from Eqs. (33) for vector ﬁelds

ην ν η ην αβ ξη aν ,aξ µ =0, aξ,p µ = δµδξ = − p ,aξ µ, p ,p µ =0. (83)

The equation for aξ emerges as ∂a ξ = [a , H ] = a , − 1 pαβp − 1 ω2aαa ∂xµ ξ P µ ξ 4 αβ 2 α µ 1 αβ 1 αβ 1 2 α 1 2 α = − 4 pαβ aξ,p µ − 4 aξ,p µ pαβ − 2 ω a [aξ,aα]µ + 2 ω [aξ,aα]µ a

β α β α =0 =0 =δµ δξ =δµ δξ 1 | {z } | {z } | {z } | {z } = − 2 pξµ.

With pξµ being skew-symmetric, we then have ∂a ∂a p = µ − ξ . (84) ξµ ∂xξ ∂xµ The equation for pην is similarly derived by ∂pξα δν = pξν , H = pξν , − 1 pαβp − 1 ω2aαa µ ∂xα P µ 4 αβ 2 α µ 1 ξν αβ 1 ξν αβ 1 2 α ξν 1 2 ξν α = − 4 pαβ p ,p µ − 4 p ,p µ pαβ − 2 ω a p ,aα µ + 2 ω p ,aα µ a

=0 =0 ν ξ ν ξ =−δµδα =−δµδα ν 2 ξ = δµω a |, {z } | {z } | {z } | {z } hence ∂pµα = ω2 aµ. (85) ∂xα The obtained equations (84) and (85) obviously agree with the canonical ﬁeld equations from above.

4.6. Canonical ﬁeld equations of a coupled Klein-Gordon-Maxwell system

The Lagrangian density LKGM of a complex Klein-Gordon ﬁeld φ that couples minimally to an electromagnetic 4-vector potential a is given by ∂φ ∂φ L = − iqaαφ + iqa φ − ω2φφ − 1 f αβf . (86) KGM ∂x ∂xα α 4 αβ α December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

36 J. Struckmeier, A. Redelbach

The components fµν of the electromagnetic field tensor are defined in Eq. (73). The conjugate fields of φ and a are obtained from the Lagrangian LKGM via ∂L ∂φ πν = KGM = − iqaν φ ∂(∂ν φ) ∂xν ∂LKGM ∂φ πν = = + iqaν φ ∂ ∂ν φ ∂xν ∂L ∂aν ∂aµ pµν = KGM = f µν ≡ − ∂(∂ν aµ) ∂xµ ∂xν

The corresponding Hamiltonian density HKGM is now emerges from LKGM by the Legendre transformation prescription

αβ ∂aα α ∂φ ∂φ HKGM = p β + π α + πα − LKGM. ∂x ∂x ∂xα

The Hamiltonian HKGM is found by replacing all partial derivatives of the ﬁelds φ and a by the respective conjugate ﬁelds, πµ and pµν ,

α α α 2 1 αβ αβ βα HKGM = παπ + iqaα π φ − φπ + ω φφ − 4 p pαβ, p = −p . (87)

Corresponding to Sect. 4.4, the derivative of the Hamiltonian density HKGM with respect to the pµν yields the canonical equation ∂a ∂H µ = KGM = − 1 p = 1 p , ∂xν ∂pµν 2 µν 2 νµ hence ∂a ∂a p = µ − ν . νµ ∂xν ∂xµ µ From the derivatives of HKGM with respect to the π and πµ, the following canonical ﬁeld equations emerge

∂φ ∂H = KGM = π − iqa φ ∂xµ ∂πµ µ µ ∂φ ∂H = KGM = πµ + iqaµφ. ∂xµ ∂πµ

The third group of canonical ﬁeld equations results from the derivatives of HKGM with respect to the aµ, and to the φ, φ as

µα ∂p ∂HKGM µ µ Def. µ − α = = iq π φ − φπ = −j ∂x ∂aµ ∂πα ∂H − = KGM = ω2φ + iqa πα ∂xα ∂φ α α ∂π ∂HKGM 2 α − = = ω φ − iqa πα. ∂xα ∂φ December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 37

Due to the property of pµν being skew-symmetric in its indices, the divergence of jµ must vanish in order for the system of canonical equations to be consistent, ∂jα ∂2pαβ = =0. ∂xα ∂xα∂xβ This requirement is indeed fulfilled, for, inserting the canonical equations, we find ∂jα ∂ = iq φπα − παφ ∂xα ∂xα ∂φ ∂πα ∂πα ∂φ = iq πα + φ − φ − πα ∂xα ∂xα ∂xα ∂xα α 2 α = iq πα − iqaαφ π − φ ω φ − iqa πα 2 α α + ω φ + iqaαπ φ − π (πα + iqaαφ) =0. By eliminating the conjugate fields pµν and πµ, the canonical field equations can be rewritten as second order partial differential equations,

2 α ∂ φ 2 2 α ∂φ ∂a α + ω − q aαa φ + 2iqaα + iqφ α =0 ∂x ∂xα ∂xα ∂x 2 α ∂ φ 2 2 α ∂φ ∂a α + ω − q aαa φ − 2iqaα − iqφ α =0, ∂x ∂xα ∂xα ∂x which coincide exactly with those that follow directly from the Euler-Lagrange equa- µ tions for the Lagrangian density LKGM. With the 4-current density j expressed in terms of φ, φ and their derivatives, ∂φ ∂φ jµ = iq φ − iqaµφ − + iqaµφ φ ∂x ∂x µ µ ∂φ ∂φ = iq φ − φ +2q2φφaµ, ∂x ∂x µ µ we equally verify that ∂jα/∂xα = 0 by inserting the second order ﬁeld equations.

4.7. The Dirac Hamiltonian density 1 The dynamics of particles with spin 2 and mass m is described by the Dirac equation. With γi, i =1,..., 4 denoting the 4×4 Dirac matrices, and ψ a four component Dirac spinor, the Dirac Lagrangian density LD is given by ∂ψ L = i ψγα − mψψ, (88) D ∂xα wherein ψ ≡ ψ†γ0 denotes the adjoint spinor of ψ. Operating with spinors and Dirac matrices, Eq. (88) is actually a shorthand for ∂ψ L = iψ γα J − mψ ψ , D K KJ ∂xα J J December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

38 J. Struckmeier, A. Redelbach

α where ψJ stands for a spinor component and γKJ for the KJ matrix element of the Dirac matrix γα. In the following we summarize some fundamental relations that µ apply for the Dirac matrices γ , and their duals, γµ, 1 1 gµν 1 = (γµγν + γν γµ) ≡ {γµ,γν } 2 2 α α γ γα = γαγ =4 1 α µ µ α µ γ γ γα = γαγ γ = −2γ α α γ γµγα = γαγµγ = −2γµ µ ν α α µ ν µν γαγ γ γ = γ γ γ γα =4 g 1 α α γαγµγν γ = γ γµγν γα =4 gµν 1 i i σµν ≡ (γµγν − γν γµ) ≡ [γµ,γν ] 2 2 det σµν =1, µ 6= ν i τ = (γ γ +3γ γ ) µν 6 µ ν ν µ αν να ν 1 τµασ = σ ταµ = δµ 1 γατ = τ γα = γ αµ µα 3i µ αµ µα µ γασ = σ γα = 3i γ 4 γατ γβ = 1 αβ 3i αβ γασ γβ = 12i 1. (89)

Herein, the symbol 1 stands for the 4 × 4 unit matrix, and the real num- µν µν bers η , ηµν ∈ R for an element of the Minkowski metric (η ) = (ηµν ) = µν diag(1, −1, −1, −1). The matrices (σ ) and (τµν ) are to be understood as 4 × 4 µν block matrices, with each block σ , τµν representing a 4 × 4 matrix of complex µν numbers. Thus, (σ ) and its inverse (τµν ) are actually 16×16 matrices of complex numbers. The Dirac Lagrangian density LD can be rendered symmetric by combining the Lagrangian density Eq. (88) with its adjoint, which leads to

i ∂ψ ∂ψ L = ψγα − γαψ − mψψ. (90) D 2 ∂xα ∂xα The resulting Euler-Lagrange equations are identical to those derived from Eq. (88), ∂ψ iγα − mψ =0 ∂xα ∂ψ i γα + mψ =0. (91) ∂xα As both Lagrangians (88) and (90) are linear in the derivatives of the ﬁelds, the December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 39

determinant of the Hessian vanishes,

∂2L det D =0. (92) "∂(∂µψ)∂ ∂ν ψ # Therefore, Legendre transformations of the Lagrangian densities (88) and (90) are irregular. Nevertheless, as a Lagrangian density is determined only up to the divergence of an arbitrary vector function F µ according to Eq. (14), one can construct ′ an equivalent Lagrangian density LD that yields identical Euler-Lagrange equations while yielding a regular Legendre transformation. The additional term21 emerges as the divergence of a vector function F µ, which may be expressed in symmetric form as i ∂ψ ∂ψ F µ = ψ σµα + σαµ ψ . 2m ∂xα ∂xα The factor m−1 was chosen to match the dimensions correctly. Explicitly, the additional term is given by ∂F β i ∂ψ ∂ψ ∂2ψ ∂2ψ ∂ψ ∂ψ = σβα + ψσβα + σαβψ + σαβ ∂xβ 2m ∂xβ ∂xα ∂xα∂xβ ∂xα∂xβ ∂xα ∂xβ i ∂ψ ∂ψ = σαβ . m ∂xα ∂xβ βα αβ Note that the double sums σ ∂β∂αψ and ∂β∂αψσ vanish identically, as we µν νµ sum over a symmetric (∂µ∂ν ψ = ∂ν ∂µψ) and a skew-symmetric (σ = −σ ) ′ factor. Following Eq. (14), the equivalent Lagrangian density is given by LD = β β LD + ∂F /∂x , which means, explicitly, i ∂ψ ∂ψ i ∂ψ ∂ψ L′ = ψγα − γαψ + σαβ − mψψ. (93) D 2 ∂xα ∂xα m ∂xα ∂xβ µν ′ Due to the skew-symmetry of the σ , the Euler-Lagrange equations (2) for LD ′ yield again the Dirac equations (91). As desired, the Hessian of LD is not singular, ∂2L′ i i det D = det σµν = , ν 6= µ, (94) " ∂ ∂µψ ∂(∂ν ψ) # m m ′ hence, the Legendre transformation of the Lagrangian density LD is now regular. It is remarkable that it is exactly a term which does not contribute to the Euler- ′ Lagrange equations that makes the Legendre transformation of LD regular and thus transfers the information on the dynamical system that is contained in the Lagrangian to the Hamiltonian description. The canonical momenta follow as ′ µ ∂LD i µ i ∂ψ αµ π = = ψγ + α σ ∂(∂µψ) 2 m ∂x ′ µ ∂LD i µ i µα ∂ψ π = = − γ ψ + σ α . (95) ∂ ∂µψ 2 m ∂x December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

40 J. Struckmeier, A. Redelbach

The Legendre transformation can now be worked out, yielding

∂ψ ∂ψ H = πν + πν − L′ D ∂xν ∂xν D i ∂ψ ∂ψ = σµν + mψψ m ∂xµ ∂xν i ∂ψ = πν − ψγν + mψψ. 2 ∂xν

As the Hamiltonian density must always be expressed in terms of the canonical momenta rather then by the velocities, we must solve Eq. (95) for ∂µψ and ∂µψ. To µ αµ this end, we multiply π by τµν — hence by the inverse of σ — from the right, µ and π by τνµ from the left,

∂ψ m i = πµ − ψγµ τ ∂xν i 2 µν ∂ψ m i = τ πµ + γµψ . (96) ∂xν i νµ 2

The Dirac Hamiltonian density is then ﬁnally obtained as

m i i H = πν − ψγν τ πµ + γµψ + mψψ. (97) D i 2 νµ 2

We may expand the products in Eq. (97) using Eqs. (89) to ﬁnd

im 1 1 2m H = ψγ πν − 3πµτ πν − πν γ ψ + ψψ. (98) D 3 2 ν µν 2 ν 3

In order to show that the Hamiltonian density HD describes the same dynamics as LD from Eq. (88), we set up the canonical equations

∂ψ ∂H 1 = D = im ψγ − πµτ ∂xν ∂πν 6 ν µν ∂ψ ∂H 1 = D = −im γ ψ + τ πν . ∂xµ ∂πµ 6 µ µν

These equations reproduce the deﬁnition of the canonical momenta from Eqs. (95) in their inverted form given by Eqs. (96). The second set of canonical equations December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 41

follows from the ψ and ψ dependence of the Hamiltonian HD, ∂πα ∂H im 2m = − D = πµγ − ψ ∂xα ∂ψ 6 µ 3 im i i ∂ψ 2m = ψγµ + σνµ γ − ψ 6 2 m ∂xν µ 3 i ∂ψ = −mψ − γν 2 ∂xν α ∂π ∂HD im µ 2m = − = − γµπ − ψ ∂xα ∂ψ 6 3 im i i ∂ψ 2m = − γ − γµψ + σµν − ψ 6 µ 2 m ∂xν 3 i ∂ψ = −mψ + γν . 2 ∂xν The divergences of the canonical momenta follow equally from the derivatives of the ﬁrst canonical equations, or, equivalently, from the derivatives of Eqs. (95),

∂πα i ∂ψ i ∂2ψ = γα + σαβ ∂xα 2 ∂xα m ∂xα∂xβ i ∂ψ = γα 2 ∂xα ∂πα i ∂ψ i ∂2ψ = − γα + σαβ ∂xα 2 ∂xα m ∂xα∂xβ i ∂ψ = − γα . 2 ∂xα The terms containing the second derivatives of ψ and ψ vanish due to the skew- symmetry of σµν . Equating ﬁnally the expressions for the divergences of the canonical momenta, we encounter, as expected, the Dirac equations (91)

i ∂ψ i ∂ψ γα = −mψ − γν 2 ∂xα 2 ∂xν i ∂ψ i ∂ψ − γα = −mψ + γν . 2 ∂xα 2 ∂xν It should be mentioned that this section is similar to the derivation of the Dirac Hamiltonian density in Ref. 22. We will show in Sect. 6.6.1 by means of a canonical ′ transformation that the transition from the Lagrangian LD (Eq. (90)) to LD corresponds in the Hamiltonian formalism to the transition from πµ, πµ to equivalent µ vectors of canonical momenta, Πµ, Π , i.e. momentum vectors of identical divergences. It will be furthermore shown in Sect. 6.6.2 that there exists a canonical transformation that interchanges the ﬁelds, ψ, ψ, with their canonical conjugates, πµ, πµ, while maintaining exactly the form of the Hamiltonian (98). ′ We ﬁnally note that the additional term in the Dirac Lagrangian density LD from Eq. (93) — as compared to the Lagrangian LD from Eq. (90) — entails December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

42 J. Struckmeier, A. Redelbach

additional terms in the energy-momentum tensor, deﬁned in Eq. (6), namely,

i Θ ν − θ ν ≡ jν (x)= ∂ ψσαν ∂ ψ + ∂ ψσνα∂ ψ − δν ∂ ψσαβ∂ ψ . µ µ µ m α µ µ α µ α β ν We easily convince ourselves by direct calculation that the divergences of Θµ and ν θµ coincide,

∂jλ i µ = ∂ ∂ ψσαλ∂ ψ + ∂ ψσαλ∂ ∂ ψ + ∂ ∂ ψσλα∂ ψ ∂xλ m λ α µ α λ µ λ µ α λα λ αβ λ αβ + ∂µψσ ∂λ∂αψ − δµ∂λ∂αψσ ∂βψ − δµ∂αψσ ∂λ∂βψ i = ∂ ψσαλ∂ ∂ ψ + ∂ ∂ ψσλα∂ ψ − ∂ ∂ ψσαβ ∂ ψ − ∂ ψσαβ ∂ ∂ ψ m α λ µ λ µ α µ α β α µ β =0,

which means that both energy-momentum tensors represent the same physical sys- ν tem. For each index µ, jµ(x) represents a conserved current vector which are all ′ associated with the transformation from LD to LD. We now derive the equations of motion for the sets of ﬁelds ψI ,ππI using the covariant Heisenberg equations (31)

∂ψ ∂πα ∂ψ ∂πα I = ψ , H , δν I = πν , H , I = ψ , H , δν I = πν , H , ∂xµ I µ µ ∂xα I µ ∂xµ I µ µ ∂xα I µ in conjunction with the fundamental Poisson bracket rules from Eqs. (33). Accord-

ing to the definition (95) of the canonical momenta of the ψI and the ψI , the fields (ψI ,π I ) and, correspondingly, ( ψI ,ππI ) constitute the pairs of canonical conjugate quantities. Again, ψI and ψI as well for the conjugate fields, πI and π I are to be regarded as independent fields. The equations for ψ and ψ follows as

∂ψ 1 1 2 = [ψ, H ] = ψ, im ψγ πα − πατ πβ − παγ ψ + m ψψ ∂xµ D µ 6 α αβ 6 α 3 µ im im 2m = ψ, ψγ πα − im ψ, πατ πβ − [ψ, παγ ψ] + ψ, ψψ 6 α µ αβ µ 6 α µ 3 µ im im = −imπατ ψ, πβ −im [ψ, πα] τ πβ − παγ [ψ, ψ] − [ψ, πα] γ ψ αβ µ µ αβ 6 α µ 6 µ α α α =0 =δµ =0 =δµ im im 2m 2m + ψγ [ψ,| π{zα] +} |ψ, ψ{z }γ πα + ψ [ψ, ψ]| +{z } ψ, ψ| {zψ } 6 α µ 6 µ α 3 µ 3 µ =0 =0 =0 =0 1 = −im τ π| α {z+ }γ ψ ,| {z } | {z } | {z } µα 6 µ December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 43

∂ψ 1 1 2 = ψ, H = ψ, im −πατ πβ − παγ ψ + ψγ πα + m ψψ ∂xµ D µ αβ 6 α 6 α 3 µ im im 2m = −im ψ, πατ πβ − ψ, παγ ψ + ψ, ψγ πα + ψ, ψψ αβ µ 6 α µ 6 α µ 3 µ im im = −imπατ ψ, πβ −im ψ, πα τ πβ − παγ ψ, ψ − ψ, πα γ ψ αβ µ µ αβ 6 α µ 6 µ α β =0 =0 =0 =δµ im im 2m 2m + ψγ |ψ, π{zα }+ |ψ, ψ{z γ} πα + ψ ψ, ψ| +{z } ψ, ψ| ψ{z } 6 α µ 6 µ α 3 µ 3 µ α =δµ =0 =0 =0 1 = im ψγ| −{zπατ} . | {z } | {z } | {z } 6 µ αµ

Similarly, the equations for the divergences of πI and πI are

∂πα 1 1 2 δν = [πν , H ] = πν , im −πατ πβ − παγ ψ + ψγ πα + m ψψ µ ∂xα D µ αβ 6 α 6 α 3 µ im im 2m = −im πν , πατ πβ − [πν , παγ ψ] + πν , ψγ πα + πν , ψψ αβ µ 6 α µ 6 α µ 3 µ im im = −imπατ πν , πβ −im [πν , πα] τ πβ − παγ [πν , ψ] − [πν , πα] γ ψ αβ µ µ αβ 6 α µ 6 µ α =0 =0 =0 =0 im im 2m 2m + ψγ [π|ν , π{zα] +} |πν ,{zψ }γ πα + ψ [πν , ψ|] {z+ } πν , |ψ {zψ } 6 α µ 6 µ α 3 µ 3 µ =0 ν =0 ν =−δµ =−δµ i | {z }2 | {z } = −δν m γ πα + ψ | {z } | {z } µ 6 α 3

∂πα 1 1 2 δν = [πν , H ] = πν , im −πατ πβ − παγ ψ + ψγ πα + m ψψ µ ∂xα D µ αβ 6 α 6 α 3 µ im im 4m = −im πν , πατ πβ − [πν , παγ ψ] − πν , ψγ πα + πν , ψψ αβ µ 6 α µ 6 α µ 3 µ im im = −imπατ πν , πβ −im [πν , πα] τ πβ − παγ [πν , ψ] − [πν , πα] γ ψ αβ µ µ αβ 6 α µ 6 µ α ν =0 =0 =−δµ =0 im im 2m 2m + ψγ [π|ν , π{zα] +} |πν ,{zψ }γ πα + ψ [πν , ψ|] {z+ } πν , |ψ {zψ } 6 α µ 6 µ α 3 µ 3 µ ν =0 =0 =−δµ =0 i 2 = δν m πα|γ {z− }ψ . | {z } | {z } | {z } µ 6 α 3 December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

44 J. Struckmeier, A. Redelbach

The complete set of equation is now ∂ψ 1 = −im τ πα + γ ψ (99) ∂xµ µα 6 µ ∂ψ 1 = −im πατ − ψγ (100) ∂xµ αµ 6 µ ∂πα i 2 = −m γ πα + ψ (101) ∂xα 6 α 3 ∂πα i 2 = m παγ − ψ . (102) ∂xα 6 α 3 In order to derive the equations for ψ and ψ, we must get rid of their actual dependence on the π and π. To this end, we solve Eqs. (99) and (100) for πµ and πµ, respectively, then calculate their divergences, and finally equate the divergences with the corresponding divergence of Eqs. (101) and (102). Thus, in the first step, we find 1 ∂ψ 1 σνµ + σνµγ ψ = −σνµτ πα = −πν im ∂xµ 6 µ µα 1 ∂ψ 1 σµν − ψγ σµν = −πατ σµν = −πν , im ∂xµ 6 µ αµ hence i ∂ψ i πν = σνµ − γν ψ (103) m ∂xµ 2 i ∂ψ i πν = σµν + ψγν . (104) m ∂xµ 2 The divergences are then ✟✟ ∂πα i ∂✟2ψ i ∂ψ = σαµ ✟ − γα ∂xα m✟✟∂xµ∂xα 2 ∂xα ✟✟ α 2 ✟ ∂π i ∂ ✟ψ µα i ∂ψ α = ✟ σ + γ . ∂xα m✟∂xµ∂xα 2 ∂xα The second derivative terms vanish due to the skew-symmetry of σνµ. In Eqs. (101) and (102), we eliminate their dependence on the canonical momentum fields by inserting Eqs. (103) and (104), respectively. This gives ∂πα im i ∂ψ i 2m = − γ σαµ − γαψ − ψ ∂xα 6 α m ∂xµ 2 3 i ∂ψ = − γα 2 ∂xα ∂πα im i ∂ψ i 2m = σµα + ψγα γ − ψ ∂xα 6 m ∂xµ 2 α 3 i ∂ψ = γα, 2 ∂xα December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 45

hence ﬁnally:

∂ψ iγα − m ψ =0 ∂xα ∂ψ i γα + m ψ =0, ∂xα which are the Dirac equations.

5. Examples of canonical transformations in covariant Hamiltonian ﬁeld theory 5.1. Point transformation

Canonical transformations for which the transformed fields ΦI only depend on on µ the original fields φI , and possibly on the independent variables x , but not on the original conjugate fields πI are referred to as point transformations. The generic form of a 4-vector generating function F 2 that defines such transformations has the components

µ µ F2 (φI ,ΠI , x)= fJ (φI , x) ΠJ . (105)

Herein, fJ = fJ (φI , x) denotes a set of diﬀerentiable but otherwise arbitrary functions. According to the general rules (19) for generating functions of type F 2, the transformed ﬁeld ΦK follows as

µ µ µ ∂F2 ∂ΠJ µ ΦK δν = ν = fJ (φI , x) ν = fJ (φI , x) δJK δν . ∂ΠK ∂ΠK

The complete set of transformation rules is then

∂f ∂f πµ = Πµ J , Φ = f (φ , x), H′ = H + Πα J . I J ∂φ K K I J ∂xα I expl

As a trivial example of a point transformation, we consider the gene rating function of the identical transformation. Deﬁning fJ (φI ) ≡ φJ in the generating function (105)

µ µ F2 (φI ,ΠI )= φJ ΠJ , (106)

the pertaining transformation rules for this particular case are

µ µ ′ πI = ΠI , ΦI = φI , H = H.

The existence of a neutral element is a necessary condition for the set of canonical transformations to form a group. December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

46 J. Struckmeier, A. Redelbach

5.2. Canonical shift of the conjugate momentum vector field πI The generator of a canonical transformation that shifts the conjugate 4-vector field πI (x) into an equivalent conjugate 4-vector field ΠI (x) can be defined in terms of a function of type F 2(φI ,ΠI , x) as

µ β µ µ ∂ ∂hJ µ ∂hJ F2 = φJ ΠJ + φJ α − δα β , (107) ∂xα ∂x ∂x ! µ µ with arbitrary diﬀerentiable x -dependent parameter functions hI (x). From the general transformation rules (22), the particular rules for this generating function are µ µ β µ ∂F2 µ ∂ ∂hI µ ∂hI πI = = ΠI + α − δα β ∂φI ∂xα ∂x ∂x ! µ µ ∂F2 µ ΦI δν = ν = φI δν ∂ΠI ∂F γ ∂3hγ ∂3hβ H′ − H = 2 = φ I − δγ I ≡ 0. ∂xγ I ∂xα∂x ∂xγ α ∂xβ∂x ∂xγ expl α α !

hence 2 α 2 µ µ µ ∂ hI ∂ hI ′ ΠI = πI + α − α , ΦI = φI , H = H. (108) ∂xµ∂x ∂xα∂x

The divergences of the ﬁelds πI and ΠI coincide since

β β 3 α 3 β β ∂ΠI ∂πI ∂ hI ∂ hI ∂πI β = β + β α − β α = β . ∂x ∂x ∂x ∂xβ∂x ∂x ∂xα∂x ∂x With regard to the canonical ﬁeld equations (5), this means that both vector ﬁelds, πI (x) and ΠI (x), emerge from the same Hamiltonian H, hence are canonically equivalent.

5.3. Local and global gauge transformation of the ﬁelds φI

A common phase transformation of the ﬁelds φI (x) of the form

iθ(x) φI (x) 7→ ΦI (x)= φI (x) e (109) is commonly called a “local gauge transformation”. We can conceive this as a point transformation that is generated by a 4-vector function of type F 2

µ µ iθ(x) F2 (φI ,ΠI , x) = ΠI e φI . (110) The pertaining transformation rules follow directly from the general rules of Eqs. (19) ∂θ(x) Φ = φ eiθ(x), Πµ = πµe−iθ(x), H′ = H + i πα φ . I I I I I ∂xα I December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 47

In the particular case that θ does not depend on the xµ, hence if θ = const., then the gauge transformation is referred to as “global”. In that case, the generating function (110) itself does no longer explicitly depend on the xµ. In contrast to the case of local gauge transformations, the Hamiltonian density is thus always iθ conserved under global gauge transformations φI (x) 7→ ΦI (x)= φI (x) e ,

iθ µ µ −iθ ′ ΦI = φI e , ΠI = πI e , H = H.

A generalization of the gauge transformation (109) of the ﬁelds φI (x) is straightforward. With SIJ (x) an invertible matrix, we may deﬁne the generating function

µ µ F2 (φI ,ΠI , x) = ΠI SIJ (x) φJ . (111)

With TIJ (x) the inverse matrix of SIJ (x), hence SIK TKJ = δIJ , the transformation rules follow as ∂S Φ = S (x) φ , Πµ = πµ T (x), H′ = H + πα T IJ φ . I IJ J I K KI K KI ∂xα J

5.4. Infinitesimal canonical transformation, generalized Noether theorem Canonical transformations were derived in Sect. 3 as the particular subset of gen- µ eral transformations of the fields φI and their conjugate momentum fields πI that preserve the action integral (11). Such a transformation depicts a symmetry transformation that is associated with a conserved four-current vector, hence with a vector whose space-time divergence vanishes. In the following, we shall work out the correlation of this conserved current with an infinitesimal canonical transfor- µ mation of the field variables. The generating function F2 of an infinitesimal transformation differs from that of an identical transformation (106) by a small quantity ǫgµ(φφ,Π, x)

µ µ µ F2 (φφ,Π, x)= φJ ΠJ + ǫg (φφ,Π, x). (112)

To ﬁrst order in ǫ, the subsequent transformation rules (19) are

∂gµ ∂gµ ∂gα Πµ = πµ − ǫ , Φ δµ = φ δµ + ǫ , H′ = H + ǫ , I I ∂φ I ν I ν ∂Πν ∂xα I I expl

µ µ µ ′ hence for δπI = πI − ΠI , δφI = φI − ΦI , and δH|CT = H − H ∂gµ ∂gµ ∂gα δπµ = ǫ , δφ δµ = −ǫ , δH| = −ǫ . (113) I ∂φ I ν ∂Πν CT ∂xα I I expl

µ As the transformation does not change the independent variables , x , all primed and unprimed quantities refer to the same space-time event x, hence δxµ = 0. With the transformation rules (113), the divergence of the four-vector of characteristic December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

48 J. Struckmeier, A. Redelbach

functions gµ is given by ∂gα ∂gα ∂φ ∂gα ∂πβ ∂gα ǫ = ǫ I + ǫ I + ǫ + O(ǫ2) ∂xα ∂φ ∂xα β ∂xα ∂xα I ∂ΠI expl α α ∂φI ∂πI 2 = δπ − δφI − δH| + O (ǫ ). I ∂xα ∂xα CT Along the system’s space-time evolution, the canonical field equations (5) apply. The derivatives of the fields with respect to the independent variables may be then replaced accordingly to yield to first order in ǫ α ∂g ∂H α ∂H ǫ α = α δπI + δφI − δH|CT. ∂x ∂πI ∂φI

If and only if the inﬁnitesimal transformation rule δH|CT for the Hamiltonian co- µ incides with the variation δH due to the variations δφI and δπI of the canonical ﬁeld variables at δxµ = 0 from Eqs. (113),

∂H ∂H α δH = δφI + α δπI , ∂φI ∂πI then this set of infinitesimal transformation rules actually defines a canonical transformation. We thus have ∂gα δH| =! δH ⇒ =0! . (114) CT ∂xα Thus, the divergence of the gµ(x) must vanish in order for the transformation (113) to be canonical, and hence to preserve the action integral (11). The gµ(x) then define a conserved four-current vector, commonly referred to as Noether current jµ(x), ∂jα(x) jµ(x) ≡ gµ(φφ,πππ, x), =0. (115) ∂xα This is the generalized Noether theorem of classical field theory in the Hamiltonian formulation. To summarize, the theorem thus states that the characteristic function gµ in the generating function (112) must have zero divergence, ∂gα/∂xα = 0, in order for the subsequent infinitesimal transformation (113) to be canonical and hence to preserve the action integral (11). In contrast to the usual derivation of this theorem in the Lagrangian formalism, we are not restricted to point transformations as the gµ may represent any divergence-free four-vector function of the canonical variables.

5.5. Canonical transformation inducing an inﬁnitesimal space-time step µ We now consider the generating function F2 of an inﬁnitesimal canonical transformation induced by the energy-momentum tensor from Eq. (7) ∂φ ∂φ θ µ = δµ H + πµ I − δµπβ I . (116) α α I ∂xα α I ∂xβ December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 49

The inﬁnitesimal canonical space-time step transformation is then generated by

µ µ µ α F2 (φφ,Π, x)= φI ΠI + θα δx . (117)

In order to illustrate this generating function, we imagine for a moment a system with only one independent variable, t. As a consequence, only one conjugate ﬁeld πI could exist for each φI . In that system, the last two terms of Eq. (116) would obviously cancel, hence, the generating function F2 would simplify to

F2(φI , ΠI ,t)= φI ΠI + H δt.

We recognize this function from point mechanics as the generator of the infinitesimal canonical transformation that shifts an arbitrary Hamiltonian system along an infinitesimal time step δt. Applying the general transformation rules (19) for generating functions of type F 2 to the generator from Eq. (117), then — similar to the preceding example — only terms of first order in δxµ need to be taken into account. As the partial derivatives µ of φI and πI are no canonical variables, these terms must be treated as explicitly ν µ x -dependent coefficients. The derivative of F2 with respect to φI yields

µ µ ∂F2 µ α µ ∂H πI = = ΠI + δx δα ∂φI ∂φI ∂πα = Πµ − δxµ I . I ∂xα

µ µ µ This means for δπI ≡ ΠI − πI ∂πα δπµ = I δxµ. (118) I ∂xα

To ﬁrst order, the general transformation rule (19) for the ﬁeld φI takes on the particular form for the actual generating function (117):

µ µ ∂F2 ΦI δν = ν ∂ΠI ∂H ∂φ ∂φ = φ δµ + δxα δµ + δµ I − δµδβ I I ν α ∂πν ν ∂xα α ν ∂xβ I ∂φ ∂φ ∂φ = φ δµ + δxα δµ I + δµ I − δµ I I ν α ∂xν ν ∂xα α ∂xν ∂φ = φ δµ + δµ I δxα, I ν ν ∂xα

hence with δφI ≡ ΦI − φI

∂φ δφ = I δxα. (119) I ∂xα December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

50 J. Struckmeier, A. Redelbach

′ The transformation rule δH|CT ≡ H −H for the Hamiltonian density ﬁnally follows from the explicit dependence of the generating function on the xµ as

∂H ∂2φ ∂2φ δH| = δxα δµ + πµ I − δµ πβ I CT α ∂xµ I ∂xα∂xµ α I ∂xβ∂xµ expl !

∂H = δxα . (120) ∂xα expl

µ The variation δH of the Hamiltonian due to the variations δφI and δπI that are induced by the canonical transformation is given by

∂H ∂H ∂H δH = δφ + δπα + δxα ∂φ I ∂πα I ∂xα I I expl

∂πβ ∂φ ∂φ ∂πβ ∂H = − I I δxα + I I δx α + δxα ∂xβ ∂xα ∂xα ∂xβ ∂xα expl

∂H = δxα. ∂xα expl

As δH|CT = H, the infinitesimal transformation generated by (117) is thus indeed canonical. Summarizing, we infer from the transformation rules (118), (119), and (120) that the generating function (117) defines the particular canonical transformation that infinitesimally shifts a given system in space-time in accordance with the canonical field equations (5). As such a canonical transformation can be re- peated an arbitrary number of times, we can induce that a transformation along finite steps in space-time is also canonical. We thus have the important result the space-time evolution of a system that is governed by a Hamiltonian density itself constitutes a canonical transformation. As canonical transformations map Hamilto- nian systems into Hamiltonian systems, it is ensured that each Hamiltonian system remains so in the course of its space-time evolution.

5.6. Lorentz gauge as a canonical point transformation of the Maxwell Hamiltonian density

The Hamiltonian density HM of the electromagnetic field was derived in Sec. 4.4. The correlation of the conjugate fields pµν with the 4-vector potential a is determined by the first field equation (78) as the generalized curl of a. This means, on the other hand, that the correlation between a and the pµν is not unique, hence that there is a gauge freedom of a. Defining a transformed vector potential A according to

∂Λ(x) A = a + , (121) µ µ ∂xµ December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 51

with Λ = Λ(x) an arbitrary diﬀerentiable function of the independent variables, we ﬁnd ∂A ∂A ∂a ∂2Λ(x) ∂a ∂2Λ(x) ∂a ∂a P = ν − µ = ν + − µ − = ν − µ = p . µν ∂xµ ∂xν ∂xµ ∂xν ∂xµ ∂xν ∂xµ∂xν ∂xµ ∂xν µν (122) We will now show that the gauge transformation (121) can be regarded as a canon- ν ical point transformation, whose generating function F2 is given by ∂ F µ(aa,PP , x)= a P αµ + (P αµΛ(x)) . (123) 2 α ∂xα In the notation of this example, the general transformation rules (19) are rewritten as ∂F µ ∂F µ ∂F α pνµ = 2 , A δµ = 2 , H′ = H + 2 , (124) ∂a ν β ∂P νβ ∂xα ν expl

which yield for the particular generating function of Eq. (123) the transformation

prescriptions

νµ ∂aα αµ ν αµ νµ p = P = δαP = P ∂aν ∂Λ(x) A δµ = a δαδµ + δαδµ ν β α ν β ν β ∂xα ∂Λ(x) ⇒ A = a + ν ν ∂xν 2 αβ ✘✘ αβ 2 ✘✘ ′ ∂ P ✘ ∂P ∂Λ(x) αβ ✘∂ ✘Λ(x) H − H = ✘✘✘ Λ(x) + + ✘P ✘ ∂xα∂xβ ∂xα ∂xβ ∂xα∂xβ ∂pαβ ∂Λ(x) ∂pαβ ∂Λ(x) = = − ∂xα ∂xβ ∂xβ ∂xα 4π ∂Λ(x) = jα(x) . (125) c ∂xα The canonical field transformation rules coincide with the correlations of Eqs. (121) and (122) that define the Lorentz gauge. Deriving the transformation rule for the Hamiltonian density, two of the three terms vanish because of the skew-symmetry of the canonical momentum tensor P νµ = −P µν . In the realm of the canonical transformation formalism of covariant Hamiltonian field theory, we must always explicitly verify that the canonical transformation rule for the Hamiltonians actually agrees with the transformation of H due to the transformation of the fields. For the Maxwell Hamiltonian HM from Eq. (77), we find 4π H = − 1 p pαβ + jα(x) a M 4 αβ c α 4π ∂Λ(x) = − 1 P P αβ + jα(x) A − 4 αβ c α ∂xα 4π ∂Λ(x) = H′ − jα(x) . M c ∂xα December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

52 J. Struckmeier, A. Redelbach

Obviously, this relation of original and transformed Maxwell Hamiltonians agrees with the canonical transformation rule (125), which means that the transformation µ generated by F2 from Eq. (123) is actually canonical. In order to determine the conserved Noether current that is associated with the canonical point transformation generated by F 2 from Eq. (123), we need the generator of the corresponding infinitesimal canonical point transformation, ∂ F µ(aa,PPP, x)= a P αµ + ǫgµ(pp, x), gµ = pαµΛ(x) . (126) 2 α ∂xα The pertaining canonical transformation rules are ∂Λ(x) ∂pαβ ∂Λ(x) P νµ = pνµ, A = a + ǫ , δH| = H′ − H = −ǫ . ν ν ∂xν CT ∂xβ ∂xα νµ νµ νµ Because of δp ≡ P − p = 0 and δaν ≡ Aν − aν , the variation δH of H due to the variation of the canonical variables simplifies to ∂H ∂pαβ ∂Λ(x) δH = δaα = −ǫ β α ∂aα ∂x ∂x and hence agrees with the corresponding infinitesimal canonical transformation

δH|CT, as required for the transformation to be canonical. The characteristic function gµ in the generating function (126) then directly yields the conserved 4-current j(x), jν ≡ gν according to Noether’s theorem from Eq. (115) ∂jβ(x) ∂ =0, jµ(x)= pαµΛ(x) . ∂xβ ∂xα

We verify that j N(x) is indeed the conserved Noether current by calculating its divergence ∂jβ(x) ∂ ∂pαβ ∂Λ = Λ+ pαβ ∂xβ ∂xβ ∂xα ∂xα ∂2pαβ ∂pαβ ∂Λ ∂pαβ ∂Λ ∂2Λ = Λ+ + + pαβ (127) ∂xα∂xβ ∂xα ∂xβ ∂xβ ∂xα ∂xα∂xβ =0. The ﬁrst and the fourth term on the right hand side of Eq. (127) vanish individually due to pνµ = −pµν . The second and the third terms cancel each other for the same reason.

5.7. Gauge transformation of the coupled Klein-Gordon-Maxwell ﬁeld, local gauge invariance

The Hamiltonian density HKGM of a complex Klein-Gordon ﬁeld that couples to an electromagnetic 4-vector potential A was introduced in Sec. 4.6 by Eq. (87). We now deﬁne for this Hamiltonian density a local gauge transformation by means of the generating function

µ 1 ∂Λ(x) F µ = Π φeiΛ(x) + φ Πµe−iΛ(x) + P αµ a + . (128) 2 α q ∂xα December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 53

In this context, the notation “local” refers to the fact that the generator (128) depends explicitly on x. The general transformation rules (19), (124) applied to the actual generating function yield for the ﬁelds 1 ∂Λ P µν = pµν , A = a + µ µ q ∂xµ Πµ = πµeiΛ(x), Φ= φeiΛ(x) (129) µ Π = πµ e−iΛ(x), Φ= φe−iΛ(x)

and for the Hamiltonian ∂Λ(x) H′ = H + i πα φ − φ πα ∂xα α α = H + iq π φ − φ π (Aα − aα) α α α α = H + iq Π Φ − Φ Π Aα − iq π φ − φ π aα. In the transformation rule for the Hamiltonian density, the term P αβ∂2Λ/∂xα∂xβ vanishes as the momentum tensor P αβ is skew-symmetric. The transformed Hamil- ′ tonian density HKGM is now obtained by inserting the transformation rules into the Hamiltonian density HKGM of Eq. (87),

′ α α α 2 1 αβ HKGM = Πα Π + iqAα Π Φ − Φ Π + ω ΦΦ − 4 P Pαβ. We observe that the Hamiltonian density (87) is form-invariant under the local canonical transformation generated by F 2 from Eq. (128). In order to derive the conserved Noether current that is associated with this symmetry transformation, we ﬁrst set up the generating function of the inﬁnitesimal canonical transformation corresponding to (129) by letting Λ → ǫΛ and expanding the exponential function up to the linear term in ǫ ǫ ∂Λ P µν = pµν , A = a + µ µ q ∂xµ Πµ = πµ 1 + iǫΛ(x) , Φ= φ 1 + iǫΛ(x) µ µ Π = π 1 − iǫΛ(x) , Φ= φ 1 − iǫΛ(x). These transformation rules are deduced from the generating function

µ ǫ ∂Λ F µ = Π φ(1 + iǫΛ)+ φ Πµ(1 − iǫΛ)+ P αµ a + 2 α q ∂xα µ ǫ ∂Λ = Π φ + φ Πµ + P αµa + iq πµφ − φ πµ Λ+ pαµ . α q ∂xα According to Noether’s theorem from Eq. (115), the expression in brackets represents the conserved Noether current jµ(x) ∂Λ jµ(x) = iq πµφ − φ πµ Λ+ pβµ . ∂xβ December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

54 J. Struckmeier, A. Redelbach

Its divergence is given by ∂jα ∂ ∂Λ ∂pβα ∂2Λ =Λ iq παφ − φπα + iq πβφ − φ πβ + + pβα. ∂xα ∂xα ∂xβ ∂xα ∂xβ∂xα (130)

With Λ(x) an arbitrary function of space-time, the divergence of jµ(x) vanishes if and only if the three terms associated with Λ(x) and its derivatives in Eq. (130) separately vanish. Regarding the canonical field equations from Sec. 4.6 that emerge α α from the Hamiltonian HKGM, we verify that indeed ∂j /∂x = 0. µ This means in particular that the first component j1 of the Noether current ∂jα jµ = iq πµφ − φ πµ , 1 =0 1 ∂xα is separately conserved, whereas the second in conjunction with the third term, ∂pαµ = jµ, pαµ = −pµα, ∂xα 1 depicts the inhomogeneous Maxwell equation which satisfies the consistency requirement ∂2pαβ ∂2pβα ∂jα = − = 1 =0. ∂xα∂xβ ∂xα∂xβ ∂xα

5.8. General local U(N) gauge transformation As an interesting example of a canonical transformation in the covariant Hamil- tonian description of classical ﬁelds, the general local U(N) gauge transformation is treated in this section. The main feature of the approach is that the terms to be added to a given Hamiltonian H in order to render it locally gauge invariant only depends on the type of ﬁelds contained in the Hamiltonian H and not on the particular form of the original Hamiltonian itself. The only precondition is that H must be invariant under the corresponding global gauge transformation, hence a transformation not depending explicitly on x.

5.8.1. External gauge ﬁeld

We consider a system consisting of a vector of N complex ﬁelds φI , I =1,...,N, and the adjoint ﬁeld vector, φ,

φ1 . φ =  .  , φ = φ1 ··· φN . φ  N    A general local linear transformation may be expressed in terms of a dimensionless † complex matrix U(x) = (uIJ (x)) and its adjoint, U that may depend explicitly on December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 55

the independent variables, xµ, as

ΦΦ= U φφ, ΦΦ= φφU † ∗ ΦI = uIJ φJ , ΦI = φJ uJI . (131)

With this notation, φI may stand for a set of I =1,...,N complex scalar fields φI or Dirac spinors. In other words, U is supposed to define an isomorphism within the space of the φI , hence to linearly map the φI into objects of the same type. The uppercase Latin letter indexes label the field or spinor number. Their transformation in iso-space are not associated with any metric. We, therefore, do not use superscripts for these indexes as there is not distinction between covariant and contravariant components. In contrast, Greek indexes are used for those components that are associated with a metric — such as the derivatives with respect to a space-time variable, xµ. As usual, summation is understood for indexes occurring in pairs. We restrict ourselves to transformations that preserve the norm φφφφ

ΦΦΦΦΦ= φφU †U φ = φφφφ =⇒ U †U = 1 = UU † ∗ ∗ ∗ ΦI ΦI = φJ uJI uIK φK = φK φK =⇒ uJI uIK = δJK = uJI uIK. This means that U † = U −1, hence that the matrix U is supposed to be unitary. The transformation (131) follows from a generating function that — corresponding to H — must be a real-valued function of the generally complex ﬁelds φ and their canonical conjugates, πµ,

µ µ µ µ † µ F2 (φφ,φφ,Π ,Π , x)= Π U φ + φφU Π µ ∗ µ = ΠK uKJ φJ + φK uKJ ΠJ . (132) According to Eqs. (19) the set of transformation rules follows as µ µ µ ∂F2 µ µ ∂F2 ∗ µ πI = = ΠK uKJ δIJ , ΦI δν = ν = φK uKJ δν δIJ ∂φI ∂ΠI µ µ µ ∂F2 ∗ µ µ ∂F2 µ πI = = δIKuKJ ΠJ , ΦI δν = ν = δν δIK uKJ φJ . ∂φI ∂ΠI The complete set of transformation rules and their inverses then read in component notation

∗ µ µ µ µ ∗ ΦI = uIJ φJ , ΦI = φJ uJI , ΠI = uIJ πJ , ΠI = πJ uJI ∗ µ ∗ µ µ µ φI = uIJ ΦJ , φI = ΦJ uJI , πI = uIJ ΠJ , πI = ΠJ uJI . (133) We assume the Hamiltonian H to be form-invariant under the global gauge transformation (131), which is given for U = const, hence for all uIJ not depending on the independent variables, xµ. In contrast, if U = U(x), the transformation (133) is referred to as a local gauge transformation. The transformation rule for the Hamil- tonian is then determined by the explicitly xµ-dependent terms of the generating December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

56 J. Struckmeier, A. Redelbach

µ function F2 according to α ∗ ∂F α ∂u ∂u H′ − H = 2 = Π IJ φ + φ IJ Πα ∂xα I ∂xα J I ∂xα J expl ∗ α ∗ ∂uIJ ∂uIJ α = π u φJ + φ uJKπ K KI ∂xα I ∂xα K ∂u = πα φ − φ πα u∗ IJ . (134) K J K J KI ∂xα In the last step, the identity ∂u∗ ∂u ∂ ∂ JI u + u∗ IK = (u∗ u )= δ =0 ∂xµ IK JI ∂xµ ∂xµ JI IK ∂xµ JK was inserted. If we want to set up a Hamiltonian H¯ that is form-invariant under the local, hence xµ-dependent transformation generated by (132), then we must compensate the additional terms (134) that emerge from the explicit xµ-dependence of the generating function (132). The only way to achieve this is to adjoin the Hamiltonian H of our system with terms that correspond to (134) with regard to their dependence on the canonical variables, φφ,φφ,ππµ,π µ. With a unitary matrix U, the uIJ -dependent terms in Eq. (134) are skew-hermitian, ∂u ∗ ∂u∗ ∂u ∂u ∗ ∂u∗ ∂u u∗ IJ = JI u = −u∗ IK , KI u∗ = u IK = − JI u∗ , KI ∂xµ ∂xµ IK JI ∂xµ ∂xµ IJ JI ∂xµ ∂xµ IK or in matrix notation ∂U † ∂U † ∂U ∂U † ∂U † ∂U U † = U = −U † , U † = U = − U †. ∂xµ ∂xµ ∂xµ ∂xµ ∂xµ ∂xµ The u-dependent terms in Eq. (134) can thus be compensated by a Hermitian matrix (aKJ ) of “4-vector gauge fields”, with each off-diagonal matrix element, aKJ ,K 6= J, a complex 4-vector field with components aKJµ, µ =0,..., 3 ∗ aKJµ = aJKµ. The number of independent gauge fields thus amount to N 2 real 4-vectors. The amended Hamiltonian H¯ thus reads ¯ α α H = H + Ha, Ha = iq πK φJ − φK πJ aKJα. (135)

With the real coupling constant q, the interaction Hamiltonian Ha is thus real. Usually, q is deﬁned to be dimensionless. We then infer the dimension of the gauge ﬁelds aKJ to be

−1 [q]=1, [aKJ ] = [L] = [m] = [∂µ]. In contrast to the given system Hamiltonian H, the amended Hamiltonian H¯ is supposed to be invariant in its form under the canonical transformation, hence

¯′ ′ ′ ′ α α H = H + Ha, Ha = iq ΠK ΦJ − ΦK ΠJ AKJα. (136) December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 57

Submitting the amended Hamiltonian H¯ from Eq. (135) to the canonical transformation generated by Eq. (132), the new Hamiltonian H¯′ emerges with Eqs. (134) and (136) as ∂F α ∂F α H¯′ = H¯ + 2 = H + H + 2 ∂xα a ∂xα expl expl

α α ∗ ∂uIJ = H + π φ J − φ π iqaKJα + u K K J KI ∂xα ! ′ α α = H + ΠK ΦJ − ΦK ΠJ iq AKJα. The original base ﬁelds, φJ , φK and their conjugates can now be expressed in terms of the transformed ones according to the rules (133), which yields, after index relabeling, the conditions µ global GT H′(ΦΦ,Φ,Πµ,Π , xµ) = H(φφ,φφ,ππµ,πµ, xµ)

α α ∂u Π Φ − Φ Πα iq A = Π Φ − Φ Πα iqu a u∗ + KI u∗ . K J K J KJα K J K J KL LIα IJ ∂xα IJ This means that the system Hamiltonian must be invariant under the global gauge transformation deﬁned by Eq. (133), whereas the gauge ﬁelds AIJµ must satisfy the transformation rule 1 ∂u A = u a u∗ + KI u∗ . (137) KJµ KL LIµ IJ iq ∂xµ IJ

We observe that for any type of canonical field variables φI and for any Hamilto- nian system H, the transformation of the 4-vector gauge fields aIJ (x) is uniquely determined according to Eq. (137) by the transformation matrix U(x) for the N fields φI . In the notation of the 4-vector gauge fields aKJ (x), K,J =1,...,N, the transformation rule is equivalently expressed as 1 ∂u A = u a u∗ + KI u∗ , KJ KL LI IJ iq ∂x IJ or, in matrix notation 1 ∂U 1 ∂U Aˆ = Uaˆ U † + U †, Aˆ = U aaUˆ † − U †, (138) µ µ iq ∂xµ iq ∂x

witha ˆµ denoting the N ×N matrices of the µ-components of the 4-vectors AIK (x), and, ﬁnally, aˆ the N × N matrix of gauge 4-vectors aIK (x). The matrix U(x) is unitary, hence belongs to the group U(N) U †(x)= U −1(x), | det U(x)| =1. For det U(x) = +1, the matrix U(x) is an element of the group SU(N). Equation (138) is the general transformation law for gauge bosons. U anda ˆµ do not commute if N > 1, hence if U is a unitary matrix rather than a complex number of modulus 1. We are then dealing with a non-Abelian gauge theory. As the matricesa ˆµ are Hermitian, the number of independent gauge 4-vectors aIK amounts December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

58 J. Struckmeier, A. Redelbach

to N real vectors on the main diagonal, and (N 2 − N)/2 independent complex oﬀ-diagonal vectors, which corresponds to a total number of N 2 independent real gauge 4-vectors for a U(N) symmetry transformation, and hence N 2 − 1 real gauge 4-vectors for a SU(N) symmetry transformation. The commutation relation of the gauge ﬁeld matrix,

α [âµ, aˆν ]IJ = aIKµ aKJν − aIKν aKJµ = aIJα C µν (139) α ⇔ [âµ, aˆν ]=âα C µν α α defines the antisymmetric structure coefficients of the gauge group, C µν = −C νµ . As a consequence of their definition (139), the coefficients satisfy the constraint:

α ξ α ξ α ξ C ξη C µν + C ξν C ηµ + C ξµ C νη =0. With

[âα, aˆµ] , aˆν =(aIKα aKNµ − aIKµ aKNα) aNJν − aIKν (aKNα aNJµ − aKNµ aNJα) =âα aˆµ aˆν − aˆµ aˆα aˆν − aˆν aˆα aˆµ +âν aˆµ aˆα, the constraint is equivalently expressed in terms of the gauge fields as the Jacobi identity for the matrix element IJ:

[âα, aˆµ] , aˆν + [âν , aˆα] , aˆµ + [âµ, aˆν ] , aˆα =0, and thus makes the group into a Lie algebra.

5.8.2. Including the gauge field dynamics With the knowledge of the required transformation rule for the gauge fields from Eq. (137), it is now possible to redefine the generating function (132) to also describe the gauge field transformation. This simultaneously defines the transformation of µν the canonical conjugates, pJK , of the gauge fields aJKµ. Furthermore, the redefined generating function yields additional terms in the transformation rule for the Hamil- tonian. 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 fields φ and the gauge field matrices aˆ (Eq. (138)) can be regarded as a canonical transformation that emerges from an explicitly xµ-dependent and µ µ real-valued generating function vector of type F2 = F2 (φφ,φφ,Π,Π,aaa,PP , x),

µ 1 ∂u F µ = Π u φ + φ u∗ Πµ + P αµ u a u∗ + KI u∗ . (142) 2 K KJ J K KJ J JK KL LIα IJ iq ∂xα IJ Accordingly, the subsequent transformation rules for canonical variables φφ,φ and their conjugates, πµ,πµ, agree with those from Eqs. (133). The rule for the gauge ﬁelds aIKα emerges as ∂F µ 1 ∂u A δµ = 2 = δµ u a u∗ + KI u∗ , KJα ν ∂P αν ν KL LIα IJ iq ∂xα IJ JK December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 59

which obviously coincides with Eq. (137), as demanded. The transformation of the momentum fields is obtained from the generating function (142) as µ αµ ∂F2 ∗ αµ pIL = = uIJ PJK uKL. (143) ∂aLIα It remains to work out the difference of the Hamiltonians that are submitted to the canonical transformation generated by (142). Hence, according to the general rule from Eq. (19), we must calculate the divergence of the explicitly xµ-dependent µ terms of F2 α ∗ ∂F α ∂u ∂u 2 = Π IJ φ + φ IJ Πα (144) ∂xα I ∂xα J I ∂xα J expl ∗ ∗ 2 ∂uKL ∂u 1 ∂uKI ∂u 1 ∂ uKI + P αβ a u∗ + u a IJ + IJ + u∗ . JK ∂xβ LIα IJ KL LIα ∂xβ iq ∂xα ∂xβ iq ∂xα∂xβ IJ We are now going to replace all uIJ -dependencies in (144) by canonical variables making use of the canonical transformation rules. The first two terms on the right- hand side of Eq. (144) can be expressed in terms of the canonical variables by means of the transformation rules (133), (137), and (143) that all follow from the generating function (142) ∗ ∗ α ∂u ∂u α ∂u ∂u Π IJ φ + φ IJ Πα = Π IJ u∗ Φ + Φ u IJ Πα I ∂xα J I ∂xα J I ∂xα JK K K KI ∂xα J α ∂u ∂u = Π IJ u∗ Φ − Φ KI u∗ Πα I ∂xα JK K K ∂xα IJ J α ∗ = iq ΠI (AIKα − uILaLJαuJK)ΦK ∗ α − iq ΦK (AKJα − uKLaLIαuIJ ) ΠJ α α α α = iq ΠK ΦJ − ΦK ΠJ AKJα − iq πK φJ − φK πJ aKJα. The second derivative term in Eq. (144) is symmetric in the indexes α and β. If we αβ (αβ) [αβ] split PJK into a symmetric PJK and a skew-symmetric part PJK in α and β αβ (αβ) [αβ] [αβ] 1 αβ βα (αβ) 1 αβ βα PJK = PJK +PJK , PJK = 2 PJK − PJK , PJK = 2 PJK + PJK , [αβ] then the second derivative term vanishes for PJK , ∂2u P [αβ] KI =0. JK ∂xα∂xβ By inserting the transformation rules for the gauge fields from Eqs. (137), the αβ remaining terms of (144) for the skew-symmetric part of PJK are converted into ∂u ∂u∗ 1 ∂u ∂u∗ P [αβ] KL a u∗ + u a IJ + KI IJ JK ∂xβ LIα IJ KL LIα ∂xβ iq ∂xα ∂xβ [αβ] [αβ] = iqpJK aKIα aIJβ − iq PJK AKIα AIJβ 1 αβ βα 1 αβ βα = 2 iq pJK − pJK aKIα aIJβ − 2 iq PJK − PJK AKIα AIJβ 1 αβ 1 αβ = 2 iqpJK (aKIα aIJβ − aKIβ aIJα) − 2 iq PJK (AKIα AIJβ − AKIβ AIJα) . December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

60 J. Struckmeier, A. Redelbach

αβ For the symmetric part of PJK , we obtain ∂u ∂u∗ 1 ∂u ∂u∗ 1 ∂2u P (αβ) KL a u∗ + u a IJ + KI IJ + KI u∗ JK ∂xβ LIα IJ KL LIα ∂xβ iq ∂xα ∂xβ iq ∂xα∂xβ IJ ∂A ∂a = P (αβ) KJα − u LIα u∗ JK ∂xβ KL ∂xβ IJ ∂A ∂A ∂a ∂a = 1 P αβ KJα + KJβ − 1 pαβ KJα + KJβ . 2 JK ∂xβ ∂xα 2 JK ∂xβ ∂xα In summary, by inserting the transformation rules into Eq. (144), the divergence of µ µ the explicitly x -dependent terms of F2 — and hence the difference of transformed and original Hamiltonians — can be expressed completely in terms of the canonical variables as α ∂F α 2 = iq Π Φ − Φ Πα A − πα φ − φ πα a ∂xα K J K J KJα K J K J KJα expl h 1 αβ 1 αβ − 2 PJK (AKIα AIJβ − AKIβ AIJα)+ 2 pJK (aKIα aIJβ − aKIβ aIJα) ∂A ∂A ∂a ∂a i + 1 P αβ KJα + KJβ − 1 pαβ KJα + KJβ . 2 JK ∂xβ ∂xα 2 JK ∂xβ ∂xα We observe that all uIJ -dependencies of Eq. (144) were expressed symmetrically in terms of the original and transformed complex scalar fields φJ , ΦJ and 4-vector gauge fields aJK ,AAJK , in conjunction with their respective canonical momenta. Consequently, a Hamiltonian of the form ¯ α α H =H(ππ,φφφ, x) + iq πK φJ − φK πJ aKJα ∂a ∂a − 1 iqpαβ (a a − a a )+ 1 pαβ KJα + KJβ 2 JK KIα IJβ KIβ IJα 2 JK ∂xβ ∂xα is then transformed according to the general rule (19) ∂F α H¯′ = H¯ + 2 ∂xα expl

into the new Hamiltonian

¯′ ′ α α H =H (ΠΠ,Φ, x) + iq ΠK ΦJ − ΦK ΠJ AKJα ∂A ∂A − 1 iq P αβ (A A − A A )+ 1 P αβ KJα − KJβ . 2 JK KIα IJβ KIβ IJα 2 JK ∂xβ ∂xα The entire transformation is thus form-conserving provided that the original Hamil- tonian H(ππ,φφφ, x) is also form-invariant if expressed in terms of the new fields, H(ΠΠ,Φ, x), according to the transformation rules (133). In other words, H(ππ,φφφ, x) must be form-invariant under the corresponding global gauge transformation. In order to completely describe the dynamics of the gauge fields aˆ(x), we must further amend the Hamiltonian by a kinetic term that describes the dynamics of the free fields aIJ , namely 1 αβ 4 pJK pKJαβ. December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 61

We must check whether this additional term is also invariant under the canonical transformation generated by Eq. (142). From the transformation rule (143), we ﬁnd

αβ ∗ αβ ∗ pJK pKJαβ = uJI PIL uLK uKM PMNαβ uNJ αβ = δNI PIL δLM PMNαβ αβ = PNL PLNαβ. (145) Thus, the total amended Hamiltonian H¯ that is form-invariant under a local U(N) symmetry transformation (131) of the ﬁelds φφ,φ is given by

H¯ =H + Hg (146) α α 1 αβ Hg =iq πK φJ − φK πJ aKJα − 4 pJK pKJαβ ∂a ∂a −1 iqpαβ (a a − a a )+ 1 pαβ KJα + KJβ . 2 JK KIα IJβ KIβ IJα 2 JK ∂xβ ∂xα We reiterate that the original Hamiltonian H must be invariant under the corresponding global gauge transformation, hence a transformation of the form of Eq. (133) with the uIK not depending on x. In the Hamiltonian description, the partial derivatives of the fields in (146) do not constitute canonical variables and must hence be regarded as xµ-dependent coefficients when setting up the canonical field equations according to Eqs. (5). µν The relation of the canonical momenta pLM to the derivatives of the fields, ν ∂aMLµ/∂x , is generally provided by the first canonical field equation (5). This means for the particular gauge-invariant Hamiltonian (146)

∂aMLµ ∂Hg ν = µν ∂x ∂pLM ∂a ∂a = − 1 p − 1 iq (a a − a a )+ 1 MLµ + MLν , 2 MLµν 2 MIµ ILν MIν ILµ 2 ∂xν ∂xµ hence ∂a ∂a p = KJν − KJµ + iq (a a − a a ) . (147) KJµν ∂xµ ∂xν KIν IJµ KIµ IJν

We observe that pKJµν occurs to be skew-symmetric in the indexes µ,ν. Here, this feature emerges from the canonical formalism and does not have to be postulated. Consequently, the value of the last term in the Hamiltonian (146) vanishes since the sum in parentheses is symmetric in α, β. As this term only contributes to the ﬁrst canonical equation, we may omit it from Hg if we simultaneously deﬁne pµν to be skew-symmetric in µ,ν. With regard to the ensuing canonical equations, the Hamiltonian Hg from Eq. (146) is then equivalent to

1 αβ α α αβ Hg = − 4 pJK pKJαβ + iq πK aKJα φJ − φK aKJα πJ − pJK aKIα aIJβ µν ! νµ pJK = −pJK. (148) December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

62 J. Struckmeier, A. Redelbach

Finally, from the locally gauge-invariant Hamiltonian (146), the canonical equation for the base ﬁelds φI is given by ∂φ ∂H¯ ∂H I = = + iqa φ ∂xµ ∂πµ ∂πµ IJµ J H¯ I I ∂φ = I + iqa φ . ∂xµ IJµ J H

This is exactly the so-called “minimum coupling rule”, which is also referred to as

the “covariant derivative”. Remarkably, in the canonical formalism this result is derived, hence does not need to be postulated.

5.9. Locally gauge-invariant Lagrangian 5.9.1. Legendre transformation for a general system Hamiltonian The equivalent gauge-invariant Lagrangian L¯ is derived by Legendre-transforming the gauge-invariant Hamiltonian H¯, deﬁned in Eqs. (146) and (148) ∂φ ∂φ ∂a L¯ = πα K + K πα + pαβ KJα − H¯, H¯ = H + H . K ∂xα ∂xα K JK ∂xβ g µν With pJK from Eq. (147) and Hg from Eq. (146), we thus have ∂a ∂a ∂a ∂a ∂a pαβ KJα − H = 1 pαβ KJα − KJβ + 1 pαβ KJα + KJβ − H JK ∂xβ g 2 JK ∂xβ ∂xα 2 JK ∂xβ ∂xα g 1 αβ 1 αβ = − 2 pJK pKJαβ − 2 iqpJK (aKIα aIJβ − aKIβ aIJα) ∂a ∂a + 1 pαβ KJα + KJβ − H 2 JK ∂xβ ∂xα g α α 1 αβ = iq πK φJ − φK πJ aKJα − 4 pJK pKJαβ. The locally gauge-invariant Lagrangian L¯ for a given system Hamiltonian H is then ∂φ ∂φ L¯ = − 1 pαβ p − iq πα φ − φ πα a + πα K + K πα − H. (149) 4 JK KJαβ K J K J KJα K ∂xα ∂xα K As implied by the Lagrangian formalism, the dynamical variables are given by both

the fields, φK , φJ , and aKJα, in conjunction with their respective partial deriva- µ tives with respect to the independent variables, x . Therefore, the pKJ in L¯ from Eq. (149) are now merely abbreviations for a combination of the Lagrangian dynamical variables. Independently of the given system Hamiltonian H, the correlation of the pKJ with the gauge fields aKJ and their derivatives is given by the first canonical equation (147).

The correlation of the momenta πI ,πI to the base ﬁelds φI , φI and their derivatives are derived from Eq. (149) for the given system Hamiltonian H via

∂H ∂φI ∂H ∂φI µ = µ − iqaIJµφJ , µ = µ + iq φJ aJIµ. (150) ∂πI ∂x ∂πI ∂x December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 63

Thus, for any globally gauge-invariant system Hamiltonian H(φI , φI ,πI ,ππI , x), the amended Lagrangian L¯ from Eq. (149) with the πI ,ππI to be determined from Eqs. (150) describes in the Lagrangian formalism the associated physical system that is invariant under local gauge transformations.

5.9.2. Klein-Gordon system Hamiltonian

The generalized Klein-Gordon Hamiltonian HKG describing N complex scalar ﬁelds φI that are associated with equal masses m is

∗ µ ∗ ∗ α 2 ∗ HKG(πµ,ππ ,φφφ,φφ )= πIαπI + m φI φI .

This Hamiltonian is clearly form-invariant under the global gauge-transformation deﬁned by Eqs. (133). Following Eqs. (146) and (148), the corresponding locally gauge-invariant Hamiltonian H¯KG is then

¯ ∗ α 2 ∗ 1 αβ HKG = πIαπI + m φI φI − 4 pJK pKJαβ ∗ α ∗ α αβ µν ! νµ + iq πK aKJα φJ − φK aKJα πJ − pJK aKIα aIJβ , pJK = −pJK . To derive the equivalent locally gauge-invariant Lagrangian L¯KG, we set up the ﬁrst canonical equation for the gauge-invariant Hamiltonian H¯KG of our actual example ¯ ∗ ¯ ∂φI ∂HKG ∂φI ∂HKG ∗ ∗ µ = ∗ µ = πIµ + iqaIJµφJ , µ = µ = πIµ − iq φJ aJIµ. ∂x ∂πI ∂x ∂πI µ ∗ µ Inserting ∂φI /∂x and ∂φI /∂x into Eq. (149), we directly encounter the locally gauge-invariant Lagrangian L¯KG as

¯ ∗ α 2 ∗ 1 αβ LKG = πIαπI − m φI φI − 4 pJK pKJαβ, with the abbreviations ∂φ ∂φ∗ π = I − iqa φ , π∗ = I + iq φ∗ a Iµ ∂xµ IJµ J Iµ ∂xµ J JIµ ∂a ∂a p = KJν − KJµ + iq (a a − a a ) . KJµν ∂xµ ∂xν KIν IJµ KIµ IJν

In explicit form, L¯KG is thus given by

∗ ∂φ ∂φ αβ L¯ = I + iq φ∗ a I − iqaα φ − m2 φ∗φ − 1 p KG ∂xα J JIα ∂x IJ J I I 4 JK KJαβ α The expressions in the parentheses represent the “minimum coupling rule,” which appears here as the transition from the kinetic momenta to the canonical momenta. By inserting L¯KG into the Euler-Lagrange equations, and H¯KG into the canonical ∗ equations, we may convince ourselves that the emerging ﬁeld equations for φI , φI , and aJK agree. This means that H¯KG and L¯KG describe the same physical system. December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

64 J. Struckmeier, A. Redelbach

5.9.3. Dirac system Hamiltonian 1 The generalized Dirac Hamiltonian (97) describing N spin- 2 ﬁelds, each of them being associated with the same mass m, i i H = −im πα − ψ γα τ πβ + γβψ + m ψ ψ , D I 2 I αβ I 2 I I I is form-invariant under global gauge transformations (133):

α i i H′ = −im Π − Ψ γα u u∗ τ Πβ + γβΨ + m Ψ u u∗ Ψ D K 2 K KI IJ αβ J 2 J K KI IJ J =δKJ =δKJ α i i = −im Π − Ψ γα |τ {z Π}β + γβΨ + m Ψ Ψ . | {z } K 2 K αβ K 2 K K K Again, the corresponding locally gauge-invariant Hamiltonian H¯D is found by adding the gauge Hamiltonian Hg from Eq. (148) i i H¯ = −im πα − ψ γα τ πβ + γβψ + m ψ ψ D I 2 I αβ I 2 I I I 1 αβ α α αβ − 4 pJK pKJαβ + iq πK ψJ − ψK πJ + pJI aIKβ aKJα.

µ µ The correlation of the canonical momenta πI , πI with the base fields ψI , ψI and their derivatives follows again from first canonical equation for H¯D ∂ψ ∂H¯ i I = D = −im τ πβ + γβψ + iqa ψ ∂xµ ∂πµ µβ I 2 I IJµ J I ∂ψ ∂H i I = D = −im πα − ψ γα τ − iq ψ a . (151) ∂xµ ∂πµ I 2 I αµ J JIµ I µ µ Inserting ∂ψI /∂x and ∂ψI /∂x into Eq. (149), we encounter the related locally gauge-invariant Lagrangian L¯D in the intermediate form 2m L¯ = − 1 pαβ p − im πατ πβ − ψ ψ , (152) D 4 JK KJαβ I αβ I 3 I I α β with the momenta πI , πI determined by Eqs. (151). We can finally eliminate the momenta of the base fields in order to express L¯D completely in Lagrangian variables. To this end, we solve Eqs. (151) for the momenta ∂ψ im −im τ πβ = I − iqa ψ + γ ψ αβ I ∂xα IKα K 6 α I i ∂ψ im πα = I + iq ψ a − ψ γ σβα. I m ∂xβ J JIβ 6 I β Then ∂ψ im σαβ ∂ψ im −m πατ πβ = I + iq ψ a − ψ γ I − iqa ψ + γ ψ . I αβ I ∂xα J JIα 6 I α m ∂xβ IKβ K 6 β I December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 65

The sums in parentheses can be regarded as a generalized “minimum coupling rule” that applies for the case of a Dirac Lagrangian. Inserting this expression into (152) yields after expanding the final form of the gauge-invariant Dirac Lagrangian ¯ ′ 1 αβ α LD = LD − 4 pJK pKJαβ + q ψK γ aKJαψJ q ∂ψ ∂ψ + ψ a σαβ J + K a σαβψ − iq ψ a a σαβψ m K KJα ∂xβ ∂xβ KJα J K KIα IJβ J i = L′ − 1 pαβ p + q ψ a − p γβ γαψ D 4 JK KJαβ K KJα 2m KJαβ J q ∂ + ψ a σαβψ . m ∂xβ K KJα J ′ LD denotes the amended system Lagrangian from Eq. (93), generalized to an N- tuple of fields ψI i ∂ψ ∂ψ 1 ∂ψ ∂ψ L′ = ψ γα I − I γαψ + I σαβ I − m ψ ψ . D 2 I ∂xα ∂xα I im ∂xα ∂xβ I I The pKJ stand for the combinations of the Lagrangian dynamical variables of the gauge fields from Eq. (147) that apply to all systems ∂a ∂a p = KJβ − KJα + iq (a a − a a ) . KJαβ ∂xα ∂xβ KIβ IJα KIα IJβ In order to set up the Euler-Lagrange equations for the locally gauge-invariant Lagrangian L¯D, we first calculate the derivatives ✟✟ ¯ ✟2 ∂ ∂LD i α ∂ψI 1 αβ ✟∂ ψI α = − γ α + σ ✟ α β ∂x ∂ ∂αψI 2 ∂x im✟ ∂x ∂x q ∂a ∂ψ − IKβ σαβψ + a σαβ K m ∂xα K IKβ ∂xα ¯ ∂LD i α ∂ψI α = γ α − mψI + qaIKαγ ψK ∂ψI 2 ∂x q ∂ψ + a σαβ K − iqa a σαβ ψ m IKα ∂xβ IJα JKβ K and ✟✟ ¯ 2 ✟ ∂ ∂LD i ∂ψI β 1 ∂ ψ✟I αβ β = β γ + ✟α β σ ∂x ∂(∂βψI ) 2 ∂x im✟∂x ∂x q ∂ψ ∂a + K a σαβ + ψ KIα σαβ m ∂xβ KIα K ∂xβ ¯ ∂LD i ∂ψI α α = − α γ − mψI + q ψK γ aKIα ∂ψI 2 ∂x q ∂ψ + K a σαβ − iq ψ a a σαβ . m ∂xβ KIα K KJα JIβ December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

66 J. Struckmeier, A. Redelbach

′ Again, the terms related to the quadratic velocity expression in LD drop out due to the skew-symmetry of σαβ. The Euler-Lagrange equations now emerge as ∂ψ q iγα I − m ψ + qa γαψ + p σαβ ψ =0 ∂xα I IKα K 2m IKαβ K ∂ψ q i I γα + m ψ − q ψ γαa − ψ σαβp =0. ∂xα I K KIα 2m K KIαβ For the case of a system with a single spinor ψ, hence for the U(1) gauge group, the locally gauge-invariant Dirac equation reduces to ∂ iq ∂A ∂A i + q A − β − α γβ γα Ψ= m Ψ. ∂xα α 4m ∂xα ∂xβ The additional term in parentheses is obviously also invariant under the U(1) gauge transformation 1 ∂Λ(x) a (x) 7→ A (x)= a (x)+ , ψ(x) 7→ Ψ(x)= ψ(x) eiΛ(x). µ µ µ q ∂xµ

5.9.4. Heisenberg equations of motion

The gauge Hamiltonian Hg from Eq. (148) converts a given globally gauge-invariant ¯ Hamiltonian system H(ψI , ψI ,ππI ,πI , x) into a locally gauge-invariant system H, where H¯ = H + Hg. We insert the Hamiltonian H¯ into the covariant Heisenberg equations (31) and apply the fundamental Poisson bracket relations from Eqs. (83) and (33). Not elaborating on terms that ﬁnally vanish by virtue of the Poisson bracket relations, the equations for ψ and ψ follows as ∂ψ I = ψ , H¯ ∂xµ I µ 1 αβ α α αβ = ψI , H− 4 pJK pKJαβ + iq πK aKJα ψJ − ψK aKJα πJ − pJK aKIα aIJβ µ h α i = [ψI , H]µ + iq [ψI , πK ]µ aKJα ψJ

α =δµ δIK

= [ψI , H]µ + iqa|IJµ{zψJ .} Similarly,

∂ψ I = ψ , H¯ ∂xµ I µ 1 αβ α α αβ = ψI , H− pJK pKJαβ + iq πK aKJα ψJ − ψK aKJα πJ − pJK aKIα aIJβ 4 µ h α i = ψI , H µ − iq ψK aKJα ψI , πJ µ

α =δµ δIJ

= ψI , H µ − iq ψK aKIµ.| {z } December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 67

The equations for the canonical momenta follow as

∂πα δν I = πν , H¯ µ ∂xα I µ ν 1 αβ α α αβ = πI , H− pJK pKJαβ + iq πK aKJα ψJ − ψK aKJα πJ − pJK aKIα aIJβ 4 µ h ν ν α i = [πI , H]µ − iq πI , ψK µ aKJα πJ

ν =−δµδIK ν ν α = [πI , H]µ + δµ|iqaIJα{z πJ}

and

∂πα δν I = πν , H¯ µ ∂xα I µ ν 1 αβ α α αβ = πI , H− pJK pKJαβ + iq πK aKJα ψJ − ψK aKJα πJ − pJK aKIα aIJβ 4 µ h ν α ν i = [πI , H]µ + iq πK aKJα [πI , ψJ ]µ

ν =−δµδIJ ν ν α = [πI , H]µ − δµ iq πK aKIα| . {z }

These equations implement the “minimum coupling rule”. Since the globally gauge- invariant Hamiltonian H by deﬁnition does not depend on the gauge ﬁelds, the equations for the aKJ and the pJK are common to all given systems H

∂a MNν = a , H¯ ∂xµ MNν µ 1 αβ α α αβ = aMNν , − 4 pJK pKJαβ + iq πK aKJα ψJ − ψK aKJα πJ − pJK aKIα aIJβ µ h 1 αβ 1 αβ αβ i = − pJKαβ aMNν ,pKJ − aMNν ,pJK pKJαβ − iq aMNν ,pJK aKIα aIJβ 4 µ 4 µ µ h i h i h i α β α β α β =δν δµ δMJ δNK =δν δµ δMK δNJ =δν δµ δMK δNJ 1 = − 2 pMNνµ|− iqa{zMIν aINµ} . | {z } | {z }

With pMNνµ being skew-symmetric in µ,ν it follows that

∂a ∂a p = MNµ − MNν + iq (a a − a a ) . MNνµ ∂xν ∂xµ MIµ INν MIν INµ December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

68 J. Struckmeier, A. Redelbach

Finally, the equation for the momenta of the gauge ﬁelds is

ξα ν ∂pNM ξν ¯ δµ = pNM , H ∂xα µ h ξν i1 αβ α α αβ = pNM , − pJK pKJαβ + iq πK aKJα ψJ − ψK aKJα πJ − pJK aKIα aIJβ 4 µ h α ξν ξν α i = iq πK pNM ,aKJα ψJ − iq ψK pNM ,aKJα πJ µ µ h i h i ξ ν ξ ν =−δαδµδNJ δMK =−δαδµδNJ δMK αβ| ξν{z } | αβ {z ξν} − iqpJK pNM ,aKIα aIJβ − iqpJK aKIα pNM ,aIJβ µ µ h i h i ξ ν ξ ν =−δαδµδMK δNI =−δβ δµδNJ δMI ν | ξ {zξ } ξα ξα | {z } = δµ iq ψM πN − πM ψN + aNJα pJM − pNJ aJMα , hence ∂pξα NM = iq ψ πξ − πξ ψ + a pξα − pξα a . ∂xα M N M N NJα JM NJ JMα 5.10. Noether current of the U(N) gauge transformation In order to determine the conserved Noether current that is associated with the SU(N) gauge transformation generated by F 2 from Eq. (142), we need the generator of the corresponding inﬁnitesimal canonical point transformation. The inﬁnitesimal

mappings of the ﬁelds φI and φI that correspond to the ﬁnite mappings of Eq. (131) are

ΦI = (δIJ + iǫuIJ ) φJ , ΦI = φJ (δJI − iǫuJI ) ,

with uIJ ∈ R denoting a matrix of arbitrary real coeﬃcients. As required, the norm φI φI is preserved under this transformation to ﬁrst order in ǫ

ΦI ΦI = φJ (δJI − iǫuJI ) (δIK + iǫuIK) φK 2 = φI φI − iǫ φJ uJI φI − φI uIK φK +O(ǫ ). ≡0 The generating function (142) is then| transposed{z into }

µ µ µ F2 = ΠK (δKJ + iǫuKJ ) φJ + φK (δKJ − iǫuKJ ) ΠJ ǫ ∂u + P αµ (δ + iǫu ) a (δ − iǫu )+ KI (δ − iǫu ) . JK KL KL LIα IJ IJ q ∂xα IJ IJ Omitting the quadratic terms in ǫ, the generating function of the sought-for in- ﬁnitesimal canonical transformation corresponding to (142) is obtained as

µ ǫ F µ = Π φ + φ Πµ + P αµ a + jµ, (153) 2 J J J J JK KJα q December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 69

with the Noether current 1 ∂u jµ = iq πµ u φ − φ u πµ + pαµ u a − a u + KJ . (154) K KJ J K KJ J JK KI IJα KIα IJ iq ∂xα To prove that jµ from Eq. (154) is indeed a conserved current, its divergence must be shown to vanish. Ordering the terms according to zeroth, first and second derivatives in the coefficients uKJ (x) yields 1 ∂jβ ∂ = u πβ φ − φ πβ + a pαβ − pαβa iq ∂xβ KJ ∂xβ K J K J JIα IK JI IKα βα ∂uKJ β β αβ αβ 1 ∂pJK + β πK φJ − φK πJ + aJIαpIK − pJI aIKα + α ∂x iq ∂x ! 1 ∂2u + KJ pαβ . iq ∂xα∂xβ JK µ With uKJ (x) arbitrary functions of space-time, the divergence of j (x) vanishes if and only if the three terms associated with uKJ (x) and its derivatives vanish separately. Regarding the canonical field equations that emerge from the locally gauge-invariant Hamiltonian (146), we verify that indeed ∂jα/∂xα = 0. µ This means in particular that the first component jJK of the Noether current ∂jα jµ = iq φ πµ − πµφ + a pαµ − pαµa , JK =0 JK J K J K JIα IK JI IKα ∂xα is separately conserved, whereas the second in conjunction with the third term, ∂pαµ JK = jµ , pαµ = −pµα , ∂xα JK JK JK depicts the inhomogeneous “QCD-Maxwell” equation which satisfies the consistency requirement ∂2pαβ ∂2pβα ∂jβ JK = − JK = JK =0. ∂xα∂xβ ∂xα∂xβ ∂xβ µ The jJK define conserved color currents, which act as sources of the color vector fields aJK . In contrast to the Abelian case of electromagnetics, the fields aJK them- selves contribute to the source terms j JK , which is referred to as the “self-coupling effect” of non-Abelian gauge theories.

5.11. Spontaneous breaking of gauge symmetry, Higgs mechanism In the previous section, we have seen that a globally gauge invariant Hamilto- µ µ µ nian system H(φI , φI , πI , πI , x ) is rendered a locally gauge invariant Hamiltonian system H¯ by amending H with Hg from Eq. (148). The obtained Hamiltonian H¯ = H + Hg contains the Hermitian matrix aIJ of 4-vector gauge fields and the matrix pIJ of their conjugates, in conjunction with the terms that couple the gauge fields aIJ to the base fields φI , φI and their respective conjugates. In this derivation, Hamiltonian Hg describes the dynamics of massless bosonic particles. As a December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

70 J. Struckmeier, A. Redelbach

ﬁrst guess, we could simply add by hand a corresponding Proca-style mass term if we wanted to describe massive gauge ﬁelds. Yet, the problem arises that the local gauge invariance of Hg then gets lost. As we now know from experiments, the vector gauge bosons W ± and the Z0 of the electroweak theory are actually massive. A way out of this dilemma is given by the “Higgs mechanism”, which will be presented now in the context of the canonical transformation approach. The key idea is to express the Hamiltonian density Hg from Eq. (148) with N = 1 in terms of a shifted potential Φ whose minimum is supposed to represent the system’s ground state

∂Φ ∂φ Φ(x)= φ(x) − ϕ, = . (155) ∂xµ ∂xµ Because of ϕ = const, the derivatives of φ with respect to the xµ must be unchanged under this transformation. As both the original as well as the transformed system are supposed to be physical, the transformation must preserve the action principle, hence must be canonical. The generating function that deﬁnes a canonical transformation with the properties of Eqs. (155) is

µ µ µ µ µ αµ F2 = Π − iq ϕa (φ − ϕ)+ φ − ϕ (Π + iqa ϕ)+ P aα. (156) µ As the transformation only affects the base fields φ and π , the gauge fields, aµ and pµν that are contained in the gauge Hamiltonian (148) must remain unchanged. The generating function (156) brings about the transformation rules

φ =Φ+ ϕ, φ = Φ+ ϕ µ πµ = Π − iq ϕaµ, πµ = Πµ + iqaµϕ νµ νµ νµ aµ = Aµ, p = P + iq η φϕ − ϕφ . As the generating function (156) does not explicitly depend on the xµ, the value of ′ the Hamiltonian is unchanged under the canonical transformation, hence Hg = Hg. ′ The transformed Hamiltonian density Hg is thus obtained by expressing the original dynamic variables of (148),

1 αβ α α µν ! νµ Hg = − 4 p pαβ + iq π φ − φ π aα, p = −p . in terms of the transformed ones,

′ 1 αβ 1 α 2 2 Hg = − 4 P Pαβ − 2 iq Φϕ − ϕΦ P α + q Φϕ − ϕΦ α α α α + iq Π AαΦ −ΦAαΠ + Π Aαϕ −ϕAαΠ 2 α +2q A Aα ϕΦ+ Φϕ +2ϕϕ . (157) We directly verify that the transformation does not change the derivatives of φ, as December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 71

required, ′ ∂Φ ∂Hg µ = µ = iq (AµΦ+ Aµϕ) ∂x H′ ∂Π g

= iqaµ (Φ+ ϕ) = iqaµφ ∂H ∂φ = g = . ∂πµ ∂xµ Hg

The original Hamiltonian Hg was shown to be form-invariant under local phase transformations φ 7→ φ exp(iθ(x)). So choosing θ(x) appropriately for φ(x) to be- come real, then Φ, ϕ, Πµ ∈ R. With this particular gauge, the transformed Hamil- tonian (157) simpliﬁes to

′ 1 αβ 2 α 2 α µν νµ Hg = − 4 P Pαβ +4q ϕA AαΦ+ ϕ A Aα , P = −P . 2 2 1 2 Now, the real quantity 4q ϕ = 2 ω defines a constant mass term pertaining to the α quadratic gauge field term A Aα. In final form, the transformed gauge Hamiltonian is thus given by

′ 1 αβ 2 α 1 2 α Hg = − 4 P Pαβ +4q ϕA AαΦ+ 2 ω A Aα. (158) The physical system that is described by the Hamiltonian density (158) emerged by means of a canonical transformation from the original density (148). Therefore, both systems are canonically equivalent. In the transformed system, we consider the Φ to be real. The corresponding degree of freedom now finds itself in the mass term of the now massive vector field Aµ. The transformation of two massive scalar fields φ = φ1 +iφ2 that interact with a massless vector field aµ into a single massive scalar field Φ1 that interacts with a now massive vector field Aµ is commonly referred to as the Higgs-Kibble mechanism.

6. General inhomogeneous local gauge transformation As a generalization of the homogeneous local U(N) gauge transformation, we now treat the corresponding inhomogeneous U(N) gauge transformation for the particular case of an N-tuple of ﬁelds φI .

6.1. External gauge ﬁelds

We again consider a system consisting of an N-tuple φ of complex ﬁelds φI with I =1,...,N, and φ its adjoint,

φ1 . φ =  .  , φ = φ1 ··· φN . φ  N    December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

72 J. Struckmeier, A. Redelbach

A general inhomogeneous linear transformation may be expressed in terms of a † complex matrix U(x) = uIJ (x) , U (x) = uIJ (x) and a vector ϕ(x) = ϕI (x) that generally depend explicitly on the independent variables, xµ, as ΦΦ= U φ + ϕϕ, ΦΦ= φφU † + ϕ

ΦI = uIJ φJ + ϕI , ΦI = φJ uJI + ϕI . (159)

With this notation, φI stands for a set of I =1,...,N complex fields φI . In other words, U is supposed to define an isomorphism within the space of the φI , hence to linearly map the φI into objects of the same type. The quantities ϕI (x) have the dimension of the base fields φI and define a local shifting transformation of the ΦI in iso-space. The transformation (159) follows from a generating function that — corresponding to H — must be a real-valued function of the generally complex fields φI and µ their canonical conjugates, πI ,

µ µ µ µ † µ F2 (φφ,φφ,Π ,Π , x)= Π U φ + ϕ + φφU + ϕ Π µ µ = ΠK uKJ φJ + ϕK + φK uKJ + ϕJ ΠJ . (160) According to Eqs. (19) the set of transformation rules follows as µ µ µ ∂F2 µ µ ∂F2 µ πI = = ΠK uKJ δJI , ΦI δν = ν = φK uKJ + ϕJ δν δJI ∂φI ∂ΠI µ µ µ ∂F2 µ µ ∂F2 µ πI = = δIK uKJ ΠJ , ΦI δν = ν = δν δIK uKJ φJ + ϕK . ∂φI ∂ΠI The complete set of transformation rules and their inverses then read in component notation

µ µ µ µ ΦI = uIJ φJ + ϕI , ΦI = φJ uJI + ϕI , ΠI = uIJ πJ , ΠI = πJ uJI µ µ µ µ φI = uIJ ΦJ − ϕJ , φI = ΦJ − ϕJ uJI , πI = uIJ ΠJ , πI = ΠJ uJI . (161) α We restrict ourselves to transformations that preserve the contraction π πα

α α † α † † Π Πα = π U U πα = π πα =⇒ U U = 1 = UU α α α ΠI ΠIα = πJ uJI uIKπKα = πK πKα =⇒ uJI uIK = δJK = uJI uIK . This means that U † = U −1, hence that the matrix U is supposed to be unitary. As a unitary matrix, U(x) is a member of the unitary group U(N) U †(x)= U −1(x), | det U(x)| =1. For det U(x) = +1, the matrix U(x) is a member of the special group SU(N). We require the Hamiltonian density H to be form-invariant under the global gauge transformation (159), which is given for U,ϕϕ = const., hence for all uIJ , ϕI not depending on the independent variables, xµ. Generally, if U = U(x), ϕ = ϕ(x), December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 73

then the transformation (161) is referred to as a local gauge transformation. The transformation rule for the Hamiltonian is then determined by the explicitly xµ- µ dependent terms of the generating function F2 according to α ∂F α ∂u ∂ϕ ∂u ∂ϕ H′ − H = 2 = Π IJ φ + I + φ IJ + J Πα ∂xα I ∂xα J ∂xα I ∂xα ∂xα J expl

α ∂uIJ ∂ϕI ∂uIJ ∂ϕJ α = π uKI φJ + + φ + uJKπ K ∂xα ∂xα I ∂xα ∂xα K ∂u ∂ϕ ∂ϕ = πα φ − φ πα u IJ + παu J + J u πα. K J K J KI ∂xα I IJ ∂xα ∂xα JI I (162) In the last step, the identity ∂u ∂u JI u + u IK =0 ∂xµ IK JI ∂xµ

was inserted. If we want to set up a Hamiltonian H1 that is form-invariant under the local, hence xµ-dependent transformation generated by (160), then we must compensate the additional terms (162) that emerge from the explicit xµ-dependence of the generating function (160). The only way to achieve this is to adjoin the Hamiltonian H of our system with terms that correspond to (162) with regard to their dependence on the canonical variables, φφ,φφ,ππµ,π µ. With a unitary matrix U, the uIJ -dependent terms in Eq. (162) are skew-hermitian, ∂u ∂u ∂u ∂u ∂u ∂u u IJ = JI u = −u IK , KI u = u IK = − JI u , KI ∂xµ ∂xµ IK JI ∂xµ ∂xµ IJ JI ∂xµ ∂xµ IK or in matrix notation ∂U † ∂U † ∂U ∂U † ∂U † ∂U U † = U = −U † , U † = U = − U †. ∂xµ ∂xµ ∂xµ ∂xµ ∂xµ ∂xµ µ The uKI ∂uIJ /∂x -dependent terms in Eq. (162) can thus be compensated by a hermitian matrix (aKJ ) of “4-vector gauge fields”, with each off-diagonal matrix element, aKJ ,K 6= J, a complex 4-vector field with components aKJµ, µ =0,..., 3 ∂u u IJ ↔ a , a = a = a∗ . KI ∂xµ KJµ KJµ KJµ JKµ µ Correspondingly, the term proportional to uIJ ∂ϕJ /∂x is compensated by the µ- components MIJ bJµ of a vector MIJ bJ of 4-vector gauge fields, ∂ϕ ∂ϕ u J ↔ M b , J u ↔ b M . IJ ∂xµ IJ Jµ ∂xµ JI Jµ IJ

The term proportional to ∂ϕJ /∂xuJI is then compensated by the adjoint vector −1 bJ MIJ . The dimension of the constant real matrix M is [M] = L and thus has the natural dimension of mass. The given system Hamiltonian H must be amended by a Hamiltonian Ha of the form α α α α H1 = H + Ha, Ha = ig πK φJ − φK πJ aKJα + πI MIJ bJα + bJαMIJ πI (163) December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

74 J. Struckmeier, A. Redelbach

in order for H1 to be form-invariant under the canonical transformation that is deﬁned by the explicitly xµ-dependent generating function from Eq. (160). With a real coupling constant g, the “gauge Hamiltonian” Ha is thus real. Submitting the amended Hamiltonian H1 to the canonical transformation generated by Eq. (160), ′ the new Hamiltonian H1 emerges as ∂F α ∂F α H′ = H + 2 = H + H + 2 1 1 ∂xα a ∂xα expl expl

α α ∂uIJ = H + π φJ − φ π igaKJα + uKI K K J ∂xα ∂ϕ ∂ϕ + πα M b + u J + b M + J u πα I IJ Jα IJ ∂xα Jα IJ ∂xα JI I ! ′ α α α α = H + ig ΠK ΦJ − ΦK ΠJ AKJα + ΠI MIJ BJα + BJαMIJ ΠI , with the AIJµ and BIµ deﬁning the gauge ﬁeld components of the transformed system. The form of the system Hamiltonian H1 is thus maintained under the canonical transformation,

′ ′ ′ ′ α α α α H1 = H + Ha, Ha = ig ΠK ΦJ − ΦK ΠJ AKJα + ΠI MIJ BJα + BJαMIJ ΠI , (164) provided that the given system Hamiltonian H is form-invariant under the corresponding global gauge transformation (161). In other words, we suppose the given system Hamiltonian H(φφ,φφ,ππµ,πµ, x) to remain form-invariant if it is expressed in terms of the transformed ﬁelds,

µ globalGT H′(ΦΦ,Φ,Πµ,Π , x) = H(φφ,φφ,ππµ,πµ, x). The gauge ﬁelds must then satisfy the condition

α α α α ig ΠK ΦJ − ΦK ΠJ AKJα + ΠI MIJ BJα + BJαMIJ ΠI ∂u = πα φ − φ πα iga + u IJ K J K J KJα KI ∂xα ∂ϕ ∂ϕ + πα M b + u J + b M + J u πα, I IJ Jα IJ ∂xα Jα IJ ∂xα JI I which yields with Eqs. (161) the following inhomogeneous transformation rules for the gauge ﬁelds aKJ , bJ , and bJ 1 ∂u A = u a u + KI u KJµ KL LIµ IJ ig ∂xµ IJ ∂ϕ B = M˜ u M b − ig A ϕ + I (165) Jµ JI IK KL Lµ IKµ K ∂xµ ∂ϕ B = b M u + ig ϕ A + I M˜ . Jµ Lµ KL KI K KIµ ∂xµ JI December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 75

Herein, M˜ denotes the inverse matrix of M, hence M˜ KJ MJI = MKJ M˜ JI = δKI . We observe that for any type of canonical field variables φI and for any Hamiltonian system H, the transformation of both the matrix aIJ as well as the vector bI of 4-vector gauge fields is uniquely determined according to Eq. (165) by the unitary matrix U(x) and the translation vector ϕ(x) that determine the local transformation of the N base fields φ. In a more concise matrix notation, Eqs. (165) are

1 ∂U A = Uaa U † + U † µ µ ig ∂xµ ∂ϕϕ MBB = UMbb − ig A ϕ + (166) µ µ µ ∂xµ ∂ϕ B M T = b M T U † + ig ϕϕAA + . µ µ µ ∂xµ

6.2. Including the gauge field dynamics With the knowledge of the required transformation rule for the gauge fields from Eq. (165), it is now possible to redefine the generating function (160) to also describe the gauge field transformation. This simultaneously defines the transformation of µν µν the canonical conjugates, pJK and qJ , of the gauge fields aJKµ and bJµ, 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 φI and the gauge fields aIJ ,bbI (Eq. (165)) can be regarded as a canonical transformation that emerges from an explicitly xµ-dependent and real-valued generating function vector of type µ µ F2 = F2 (φφ,φφ,Π,Π,aaa,PPP,bbb,bb,QQQ,QQ, x),

µ µ µ F2 = ΠK uKJ φJ + ϕK + φK uKJ + ϕJ ΠJ (167) αµ 1 ∂u + P αµ + ig M˜ Q αµϕ − ig ϕ Q M˜ u a u + KI u JK LJ L K J L LK KN NIα IJ ig ∂xα IJ αµ ∂ϕ ∂ϕ + Q M˜ u M b + K + b M u + J M˜ Qαµ. L LK KI IJ Jα ∂xα Kα IK IJ ∂xα LJ L

With the ﬁrst line of (167) matching Eq. (160), the transformation rules for canonical variables φφ,φ and their conjugates, πµ,πµ, agree with those from Eqs. (161). December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

76 J. Struckmeier, A. Redelbach

The rule for the gauge fields AKJα, BKα, and BKα emerge as ∂F µ 1 ∂u A δµ = 2 = δµ u a u + KI u KJα ν ∂P αν ν KN NIα IJ ig ∂xα IJ JK ∂F µ ∂ϕ ∂u B δµ = 2 = δµM˜ u M b + K − igu a u + KI u ϕ Lα ν αν ν LK KI IJ Jα ∂xα KN NIα IJ ∂xα IJ J ∂QL ∂ϕ = δµM˜ u M b + K − ig A ϕ ν LK KI IJ Jα ∂xα KJα J ∂F µ ∂ϕ ∂u B δµ = 2 = δµ b M u + J + ϕ igu a u + KI u M˜ Lα ν ∂Qαν ν Kα IK IJ ∂xα K KN NIα IJ ∂xα IJ LJ L ∂ϕ = δµ b M u + J + ig ϕ A M˜ , ν Kα IK IJ ∂xα K KJα LJ which obviously coincide with Eqs. (165) as the generating function (167) was de- vised accordingly. The transformation of the conjugate momentum fields is obtained from the generating function (167) as µ νµ ∂F2 ˜ νµ ˜ νµ ˜ νµ qJ = = MIJ uIK MLK QL , MKJ QK = uJI MKI qK ∂bJν µ νµ ∂F2 νµ ˜ νµ ˜ νµ ˜ qJ = = QL MLK uKI MIJ , QK MKJ = qK MKI uIJ (168) ∂bJν µ νµ ∂F2 νµ ˜ νµ νµ ˜ pIN = = uIJ PJK + ig MLJ QL ϕK − ig ϕJ QL MLK uKN ∂aNIν νµ ˜ νµ νµ ˜ = uIJ PJK + ig MLJ QL ΦK − ig ΦJ QL MLK uKN ˜ νµ νµ ˜ − ig MLI qL φN + ig φI qL MLN . Thus, the expression νµ ˜ νµ νµ ˜ pIN + ig MLI qL φN − ig φI qL MLN νµ ˜ νµ νµ ˜ = uIJ PJK + ig MLJ QL ΦK − ig ΦJ QL MLK uKN (169) transforms homogeneously under the gauge transformation generated by Eq. (167). Making use of the initially defined mapping of the base fields (159), the transformation rule (165) for the gauge fields bK is converted into ∂Φ ∂φ J − ig A Φ − M B = u L − iga φ − M b . (170) ∂xµ JKµ K JK Kµ JL ∂xµ LKµ K LK Kµ The above transformation rules can also be expressed more clearly in matrix notation qνµ = M T U †M˜ T Qνµ, M˜ T Qνµ = UM˜ T qνµ νµ νµ qνµ = Q MUM,˜ Q M˜ = qνµMU˜ † (171) νµ pνµ = U † P νµ + ig M˜ T Qνµ ⊗ ϕ − ig ϕ ⊗ Q M˜ U December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 77

and ∂Φ ∂φφ − ig A Φ − MBB = U − igaa φ − Mbb ∂xµ µ µ ∂xµ µ µ νµ P νµ +ig M˜ T Qνµ ⊗ Φ −ig Φ ⊗ Q M˜ =U pνµ +ig M˜ T qνµ ⊗ φ − ig φ ⊗ qνµM˜ U †. Equation (170) can be regarded as an “extended minimum coupling rule,” with the respective third terms arising from the inhomogeneous part of the gauge transformation. It remains to work out the diﬀerence of the Hamiltonians that are submitted to the canonical transformation generated by (167). Hence, according to the general rule from Eq. (19), we must calculate the divergence of the explicitly xµ-dependent µ terms of F2 α ∂F α ∂u ∂ϕ ∂u ∂ϕ 2 = Π KJ φ + K + φ KJ + J Πα ∂xα K ∂xα J ∂xα K ∂xα ∂xα J expl

αβ ˜ αβ αβ ˜ + PJK + ig MLJ QL ϕK − ig ϕJ QL MLK ∂u ∂u 1 ∂u ∂u 1 ∂2u · KN a u + u a IJ + KI IJ + KI u ∂xβ NIα IJ KN NIα ∂xβ ig ∂xα ∂xβ ig ∂xα∂xβ IJ ∂ϕ ∂ϕ αβ ∂u + M˜ Qαβ K − J Q M˜ igu a u + KI u LJ L ∂xβ ∂xβ L LK KN NIα IJ ∂xα IJ 2 2 αβ ∂u ∂ ϕ ∂u ∂ ϕ + Q M˜ KI M b + K + b M IJ + J M˜ Qαβ. L LK ∂xβ IJ Jα ∂xα∂xβ Kα IK ∂xβ ∂xα∂xβ LJ L (172)

We are now going to replace all uIJ - and ϕK -dependencies in (172) by canonical variables making use of the canonical transformation rules. To this end, the terms of Eq. (172) are split into three blocks. The ΠΠ-dependent terms of can be converted this way by means of the transformation rules (161) and (165)

α ∂u ∂ϕ ∂u ∂ϕ Π KJ φ + K + φ KJ + J Πα K ∂xα J ∂xα K ∂xα ∂xα J α ∂u ∂ϕ ∂u ∂ϕ = Π KJ u (Φ − ϕ )+ K + Φ − ϕ u KJ + J Πα K ∂xα JI I I ∂xα I I IK ∂xα ∂xα J α α α α = ig ΠK ΦJ − ΦK ΠJ AKJα + ΠK MKJ BJα + BKαMJK ΠJ α α α α − ig πK φJ − φK πJ aKJα − πK MKJ bJα + bKαMJK πJ . (173) The second derivative terms in Eq. (172) are symmetric in the indices α and β. If αβ αβ (αβ) (αβ) we split PJK and QJ into a symmetric PJK ,QJ and a skew-symmetric parts [αβ] [αβ] PJK , PJ in α and β

αβ (αβ) [αβ] [αβ] 1 αβ βα (αβ) 1 αβ βα PJK = PJK + PJK , PJK = 2 PJK − PJK , PJK = 2 PJK + PJK

αβ (αβ) [αβ] [αβ] 1 αβ βα (αβ) 1 αβ βα QJ = QJ + QJ ,QJ = 2 QJ − QJ ,QJ = 2 QJ + QJ , December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

78 J. Struckmeier, A. Redelbach

[αβ] [αβ] then the second derivative terms in Eq. (172) vanish for PJK and QJ ,

2 2 2 ∂ u ∂ ϕ [αβ] ∂ ϕ P [αβ] KI =0, J Q[αβ] =0, Q K =0. JK ∂xα∂xβ ∂xα∂xβ J K ∂xα∂xβ

By inserting the transformation rules for the gauge ﬁelds from Eqs. (165), the αβ remaining terms of (172) for the skew-symmetric part of PJK are converted into

[αβ] ˜ [αβ] [αβ] ˜ PJK + ig MLJ QL ϕK − ig ϕJ QL MLK ∂u ∂u 1 ∂u ∂u · KN a u + u a IJ + KI IJ ∂xβ NIα IJ KN NIα ∂xβ ig ∂xα ∂xβ ∂ϕ ∂ϕ [αβ] + M˜ Q[αβ] K − J Q M˜ ig A LJ L ∂xβ ∂xβ L LK KJα [αβ] ∂u ∂u + Q M˜ KI M b + b M IK M˜ Q[αβ] L LK ∂xβ IJ Jα Jα IJ ∂xβ LK L 1 αβ = − 2 ig PJK (AKIαAIJβ − AKIβAIJα) 1 ˜ ˜ αβ + 2 ig BJβMKJ AKIαMIL − BJαMKJ AKIβMIL QL 1 αβ ˜ ˜ − 2 ig QL MLI AIKαMKJ BJβ − MLI AIKβMKJ BJα 1 αβ + 2 igpJK (aKIαaIJβ − aKIβaIJα) 1 ˜ ˜ αβ − 2 ig bJβMKJ aKIαMIL − bJαMKJ aKIβMLI qL 1 αβ ˜ ˜ + 2 ig qL MLI aIKαMKJ bJβ − MLI aIKβMKJ bJα . (174) αβ αβ For the symmetric parts of PJK and QJ , we obtain

(αβ) ˜ (αβ) (αβ) ˜ PJK + ig MLJ QL ϕK − ig ϕJ QL MLK ∂u ∂u 1 ∂u ∂u 1 ∂2u · KN a u + u a IJ + KI IJ + KI u ∂xβ NIα IJ KL LIα ∂xβ ig ∂xα ∂xβ ig ∂xα∂xβ IJ ∂ϕ ∂ϕ (αβ) + M˜ Q(αβ) K − J Q M˜ ig A LJ L ∂xβ ∂xβ L LK KJα 2 2 (αβ) ∂u ∂ ϕ ∂u ∂ ϕ + Q M˜ KI M b + K + b M IK + K M˜ Q(αβ) L LK ∂xβ IJ Jα ∂xα∂xβ Jα IJ ∂xβ ∂xα∂xβ LK L (αβ) ∂A ∂a = P (αβ) + ig M˜ Q(αβ) ϕ − ig ϕ Q M˜ KJα − u LIα u JK LJ L K J L LK ∂xβ KL ∂xβ IJ 2 (αβ) ∂u ∂ ϕ ∂ϕ + Q M˜ KI M b + K − ig A J L LK ∂xβ IJ Jα ∂xα∂xβ KJα ∂xβ ∂u ∂2ϕ ∂ϕ + b M IK + K + ig J A M˜ Q(αβ) Jα IJ ∂xβ ∂xα∂xβ ∂xβ JKα LK L December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 79

∂A ∂A αβ ∂B ∂B ∂B ∂B = 1 P αβ KJα + KJβ + 1 Q Kα + Kβ + 1 Kα + Kβ Qαβ 2 JK ∂xβ ∂xα 2 K ∂xβ ∂xα 2 ∂xβ ∂xα K ∂a ∂a ∂b ∂b ∂b ∂b − 1 pαβ KJα + KJβ − 1 qαβ Kα + Kβ − 1 Kα + Kβ qαβ. 2 JK ∂xβ ∂xα 2 K ∂xβ ∂xα 2 ∂xβ ∂xα K (175) In summary, by inserting the transformation rules into Eq. (172), the divergence of µ µ the explicitly x -dependent terms of F2 — and hence the diﬀerence of transformed and original Hamiltonians — can be expressed completely in terms of the canonical variables as α ∂F α α 2 = ig Π Φ − Φ Πα A + Π M B + B M Πα ∂xα K J K J KJα K KJ Jα Kα JK J expl − ig πα φ − φ πα a − πα M b + b M πα K J K J KJα K KJ Jα Kα JK J 1 αβ 1 αβ − 2 ig PJK (AKIαAIJβ − AKIβAIJα)+ 2 igpJK (aKIαaIJβ− aKIβaIJα) 1 ˜ ˜ αβ + 2 ig BJβMKJ AKIαMIL − BJαMKJ AKIβMLI QL 1 αβ ˜ ˜ − 2 ig QL MLI AIKαMKJ BJβ − MLI AIKβMKJ BJα 1 ˜ ˜ αβ − 2 ig bJβMKJ aKIαMIL − bJαMKJ aKIβMLI qL 1 αβ ˜ ˜ + 2 ig qL MLI aIKαMKJ bJβ − MLI aIKβMKJ bJα

∂A ∂A αβ ∂B ∂B ∂B ∂B + 1 P αβ KJα + KJβ + 1 Q Kα + Kβ + 1 Kα + Kβ Qαβ 2 JK ∂xβ ∂xα 2 K ∂xβ ∂xα 2 ∂xβ ∂xα K ∂a ∂a ∂b ∂b ∂b ∂b − 1 pαβ KJα + KJβ − 1 qαβ Kα + Kβ − 1 Kα + Kβ qαβ. 2 JK ∂xβ ∂xα 2 K ∂xβ ∂xα 2 ∂xβ ∂xα K We observe that all uIJ -dependencies of Eq. (172) were expressed symmetrically in terms of both the original and the transformed complex base ﬁelds φJ , ΦJ and 4- vector gauge ﬁelds aJK ,AAJK ,bJ ,BBJ , in conjunction with their respective canonical momenta. Consequently, an amended Hamiltonian H2 of the form α α α α H2 = H(ππ,φφφ, x) + ig πK φJ − φK πJ aKJα + πK MKJ bJα + bKαMJK πJ ∂a ∂a − 1 igp αβ (a a − a a )+ 1 pαβ KJα + KJβ 2 JK KIα IJβ KIβ IJα 2 JK ∂xβ ∂xα 1 ˜ αβ + 2 ig bJβMKJ aKIα − bJαMKJ aKIβ MLI qL 1 αβ ˜ − 2 ig qL MLI aIKαMKJ bJβ − aIKβM KJ bJα ∂b ∂b ∂b ∂b + 1 qαβ Kα+ Kβ + 1 Kα + Kβ qαβ (176) 2 K ∂xβ ∂xα 2 ∂xβ ∂xα K is then transformed according to the general rule (19) ∂F α H′ = H + 2 2 2 ∂xα expl

December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

80 J. Struckmeier, A. Redelbach

into the new Hamiltonian ′ α α α α H2 = H(ΠΠ,Φ, x) + ig ΠK ΦJ − ΦK ΠJ AKJα + ΠK MKJ BJα + BKαMJK ΠJ ∂A ∂A − 1 ig P αβ (A A − A A )+ 1 P αβ KJα + KJβ 2 JK KIα IJβ KIβ IJα 2 JK ∂xβ ∂xα 1 ˜ αβ + 2 ig BJβMKJ AKIα − BJαMKJ AKIβ MLI QL 1 αβ ˜ − 2 ig QL MLI AIKαMKJ BJβ − AIKβM KJ BJα αβ ∂B ∂B ∂B ∂B + 1 Q Kα + Kβ + 1 Kα + Kβ Qαβ. (177) 2 K ∂xβ ∂xα 2 ∂xβ ∂xα K The entire transformation is thus form-conserving provided that the original Hamil- tonian H(ππ,φφφ, x) is also form-invariant if expressed in terms of the new fields, H(ΠΠ,Φ, x) = H(ππ,φφφ, x), according to the transformation rules (161). In other words, H(ππ,φφφ, 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, H3 = H2 + Hdyn and H3 = H2 + Hdyn. In other words, ′ Hdyn and the form-alike Hdyn must be chosen in a way that the transformation ′ properties of the canonical equations for the gauge fields emerging from H3 and H3 are compatible with the canonical transformation rules (165). These requirements ′ uniquely determine the form of both Hdyn and Hdyn. Thus, the Hamiltonians (176) ′ and (177) must be further amended by terms Hdyn and Hdyn that describe the dynamics of the free 4-vector gauge fields, aKJ ,bbJ and AKJ ,BBJ , respectively. Of course, Hdyn 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 fulfills this requirement is obtained from Eqs. (168) and (169) 1 αβ 1 αβ ˜ αβ αβ ˜ Hdyn = − 2 qJ qJαβ − 4 pIJ + ig MLI qL φJ − ig φI qL MLJ ˜ ˜ · pJIαβ + ig MKJ qKαβ φI − ig φJ qKαβMKI . (178) The condition for the first term to be form-invariant is αβ αβ ˜ ˜ qJ qJαβ = QL MLK uKI MIJ MNJ uNR MSR QSαβ

! 2 =δIN (det M) 2 αβ |˜ {z˜ } = (det M) QL MLK MJK QJαβ

! −2 =δLJ (det M) αβ | {z } = QJ QJαβ The mass matrix M must thus be orthogonal MM T = 1 (det M)2. (179) December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 81

′ From H3 and, correspondingly, from H3, we will work out the condition for the canonical ﬁeld equations to be consistent with the canonical transformation rules (165) for the gauge ﬁelds and their conjugates (168). With Hdyn from Eq. (178), the total amended Hamiltonian H3 is now given by

H3 = H2 + Hdyn = H + Hg (180) α α 1 αβ Hg = ig πK φJ − φK πJ aKJα − 2 igpKJ aJIα aIKβ − aJIβ aIKα ∂a ∂a ∂b ∂b ∂b ∂b + 1 pαβ JKα + JKβ + 1 qαβ Jα + Jβ + 1 Jα + Jβ qαβ 2 KJ ∂xβ ∂xα 2 J ∂xβ ∂xα 2 ∂xβ ∂xα J α α 1 ˜ αβ + πK MKJ bJα + bKαMJKπJ + 2 ig bJβMKJ aKIα − bJαMKJ aKIβ MLI qL 1 αβ ˜ 1 αβ − 2 ig qL MLI aIKαMKJ bJβ − aIKβ MKJ bJα − 2 qJ qJαβ 1 αβ ˜ αβ αβ ˜ − 4 pIJ + ig M LI qL φJ − ig φI qL MLJ ˜ ˜ · pJIαβ + ig MKJ qKαβ φI − ig φJ qKαβMKI . We reiterate that the system Hamiltonian H must be invariant under the corresponding global gauge transformation, hence a transformation of the form of Eq. (161) with the uIK not depending on x. In the Hamiltonian description, the partial derivatives of the fields in (180) do not constitute canonical variables and must hence be regarded as xµ-dependent coefficients when setting up the canonical field equations. The relation of the canonical µν ν momenta pNM to the derivatives of the fields, ∂aMNµ/∂x , is generally provided by the first canonical field equation (5). This means for the particular Hamilto- nian (180)

∂aMNµ ∂Hg ν = µν ∂x ∂pNM ∂a ∂a = − 1 ig (a a − a a )+ 1 MNµ + MNν 2 MIµ INν MIν INµ 2 ∂xν ∂xµ 1 1 ˜ ˜ − 2 pMNµν − 2 ig MIM qIµν φN − φM qIµν MIN , hence ∂a ∂a p = KJν − KJµ KJµν ∂xµ ∂xν ˜ ˜ + ig aKIν aIJµ − aKIµ aIJν − MIKqIµν φJ + φK qIµν MIJ . (181) Rewriting Eq. (181) in the form ∂a ∂a p +igM˜ q φ −igφ q M˜ = KJν − KJµ +ig(a a −a a ), KJµν IK Iµν J K Iµν IJ ∂xµ ∂xν KIν IJµ KIµ IJν we realize that the left-hand side transforms homogeneously according to Eq. (169). On the basis of the transformation rule for the gauge ﬁelds aµ from Eqs. (166), it is easy to verify that the right-hand side follows the same homogeneous transformation December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

82 J. Struckmeier, A. Redelbach

rule. The canonical equation (181) is thus generally consistent with the canonical transformation rules.

The corresponding reasoning applies for the canonical momenta qJµν and qJµν

∂bNµ ∂Hg 1 1 ˜ ν = µν = − 2 qNµν − 2 ig MNI (aIKµMKJ bJν − aIKν MKJ bJµ) ∂x ∂qN ∂b ∂b + 1 Nµ + Nν + 1 ig M˜ p + ig M˜ q φ − ig φ q M˜ φ 2 ∂xν ∂xµ 2 NI IJµν KI Kµν J I Kµν KJ J ∂bNµ ∂Hg 1 1 ˜ ν = µν = − 2 qNµν + 2 ig bJν MKJ aKIµ − bJµMKJ aKIν MNI ∂x ∂qN ∂b ∂b + 1 Nµ + Nν − 1 ig φ p + ig M˜ q φ − ig φ q M˜ M˜ , 2 ∂xν ∂xµ 2 J JIµν KJ Kµν I J Kµν KI NI hence with the canonical equation (181) ∂b ∂b q = Jν − Jµ + ig M˜ (a M b − a M b ) Jµν ∂xµ ∂xν JI IKν KL Lµ IKµ KL Lν ∂a ∂a + ig M˜ IKν − IKµ + ig (a a − a a ) φ JI ∂xµ ∂xν ILν LKµ ILµ LKν K ∂b ∂b q = Jν − Jµ − ig b M a − b M a M˜ Jµν ∂xµ ∂xν Lµ KL KIν Lν KL KIµ JI ∂a ∂a − ig φ KIν − KIµ + ig (a a − a a ) M˜ . (182) K ∂xµ ∂xν KLν LIµ KLµ LIν JI 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 ∂MBB ∂MBB MQQ = ν − µ + ig (A MBB − A MBB ) µν ∂xµ ∂xν ν µ µ ν ∂AA ∂AA + ig ν − µ + ig (A A − A A ) Φ. ∂xµ ∂xν ν µ µ ν Applying now the transformation rules for the gauge fields Aν ,BBµ from Eqs. (166), and for the base fields Φ from Eqs. (159), we find ∂Mbb ∂Mbb MQQ = U ν − µ + ig (a Mbb − a Mbb ) µν ∂xµ ∂xν ν µ µ ν ∂aa ∂aa + ig ν − µ + ig (a a − a a ) φ ∂xµ ∂xν ν µ µ ν = UM qµν . The canonical equations (182) are thus compatible with the canonical transformation rules (171) provided that M M˜ T = . (det M)2 December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 83

Thus, the mass matrix M must be orthogonal. This restriction was already encoun- tered with Eq. (179).

We observe that both pKJµν and qJµν , qJµν 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 Hamil- tonian (180) that are symmetric in µ,ν must vanish. As these terms only contribute to the ﬁrst canonical equations, we may omit them from Hg if we simultaneously deﬁne pJKµν and qJµν to be skew-symmetric in µ,ν. With regard to the ensuing canonical equations, the gauge Hamiltonian Hg from Eq. (180) is then equivalent to

β β αβ 1 αβ Hg = ig πK φJ − φK πJ aKJβ − igpJI aIKα aKJβ − 2 qJ qJαβ β αβ ˜ β ˜ αβ + πK − ig qL MLI aIKα MKJ bJβ + bKβMJK πJ + igaJIαMLI qL 1 αβ ˜ αβ αβ ˜ − 4 pIJ + ig MLI qL φJ − ig φI qL MLJ ˜ ˜ · pJIαβ + ig MKJ qKαβ φI − ig φJ qKαβMKI

µν ! νµ µν ! νµ pJK = −pJK , qJ = −qJ . (183)

Setting the mass matrix M to zero, Hg reduces to the gauge Hamiltonian of the homogeneous U(N) gauge theory (Struckmeier and Reichau 2012). The other terms describe the dynamics of the 4-vector gauge ﬁelds bJ . From the locally gauge- invariant Hamiltonian (180), the canonical equations for the base ﬁelds φI , φI are given by

∂φ ∂H ∂H I = 3 = + iga φ + M b ∂xµ ∂πµ ∂πµ IJµ J IJ Jµ H3 I I

∂φ ∂H ∂H I = 3 = − ig φ a + b M . (184) ∂xµ ∂πµ ∂πµ J JIµ Jµ IJ H3 I I

These equations represent the generalized “minimum coupling rules” for our particular case of a system of two sets of gauge ﬁelds, aJK and bJ . The canonical ﬁeld equation from the bJ ,bJ dependencies of Hg follow as

µα ∂qK ∂Hg µ ˜ αµ α = − = −MJK πJ + igaJIαMLI qL ∂x ∂bKµ µα ∂qJ ∂Hg µ αµ ˜ α = − = −πK + ig qL MLI aIKα MKJ . ∂x ∂bJµ α α Inserting πJ , πJ as obtained from Eqs. (184) for a particular system Hamiltonian H, α α terms proportional to bI and bI emerge with no other dynamical variables involved. Such terms describe the masses of bosons that are associated with the gauge ﬁelds bI . December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

84 J. Struckmeier, A. Redelbach

6.3. Gauge-invariant Lagrangian

As the system Hamiltonian H does not depend on the gauge ﬁelds aKJ and bJ , the gauge Lagrangian Lg that is equivalent to the gauge Hamiltonian Hg from Eq. (180) is derived by means of the Legendre transformation

∂a ∂b ∂b L = pαβ KJα + qαβ Jα + Jα qαβ − H , g JK ∂xβ J ∂xβ ∂xβ J g µν µν µν with pJK from Eq. (181) and qJ , qJ from Eqs. (182). We thus have ∂a ∂a ∂a ∂a ∂a pαβ KJα = 1 pαβ KJα − KJβ + 1 pαβ KJα + KJβ JK ∂xβ 2 JK ∂xβ ∂xα 2 JK ∂xβ ∂xα ∂a ∂a = − 1 pαβ p + 1 pαβ KJα + KJβ 2 JK KJαβ 2 JK ∂xβ ∂xα 1 αβ ˜ ˜ − 2 igpJK aKIα aIJβ − aKIβ aIJα − MIK qIβα φJ + φK qIβαMIJ , and, similarly ∂b qαβ Jα = − 1 qαβ q − 1 ig qαβM˜ (a M b − a M b ) J ∂xβ 2 J Jαβ 2 J JI IKα KL Lβ IKβ KL Lα 1 αβ ˜ ˜ ˜ + 2 ig qJ MJI pILαβ + ig MKI qKαβ φL − ig φI qKαβMKL φL ∂b ∂b + 1 qαβ Jα + Jβ 2 J ∂xβ ∂xα ∂b Jα qαβ = − 1 qαβ q + 1 ig b M a − b M a M˜ qαβ ∂xβ J 2 J Jαβ 2 Lβ KL KIα Lα KL KIβ JI J 1 ˜ ˜ ˜ αβ − 2 ig φI pILαβ + ig MKI qKαβ φL − ig φI qKαβMKL MJLqJ ∂b ∂b + 1 Jα + Jβ qαβ. 2 ∂xβ ∂xα J

With the gauge Hamiltonian Hg from Eq. (180), the gauge Lagrangian Lg is then

1 αβ α α Lg = − 2 qJ qJαβ − πK (igaKJαφJ + MKJ bJα)+ ig φK aKJα − bKαMJK πJ 1 αβ ˜ αβ αβ ˜ − 4 pIJ + ig MLI qL φJ − ig φI qL MLJ ˜ ˜ · pJIαβ + ig MKJ qKαβ φI − ig φJ qKαβMKI According to Eq. (181), the last product can equivalently be expressed as 1 αβ − 4 fIJ fJIαβ, with ∂a ∂a f = JIβ − JIα + ig (a a − a a ) . (185) JIαβ ∂xα ∂xβ JKβ KIα JKα KIβ

With regard to canonical variables πK ,ππK , Lg is still a Hamiltonian. The ﬁnal total gauge-invariant Lagrangian L3 for the given system Hamiltonian H then emerges December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 85

from the Legendre transformation

∂φ ∂φ L = L + πα J + J πα − H(φ , φ ,π ,ππ , x) 3 g J ∂xα ∂xα J I I I I ∂φ ∂φ = πα J − iga φ − M b + J + ig φ a − b M πα J ∂xα JKα K JK Kα ∂xα K KJα Kα JK J 1 αβ 1 αβ − 4 fIJ fJIαβ − 2 qJ qJαβ − H. (186)

As implied by the Lagrangian formalism, the dynamical variables are given by

both the ﬁelds, φI , φI , aKJ , bJ , and bJ , and their respective partial derivatives µ with respect to the independent variables, x . Therefore, the momenta qJ and qJ of the Hamiltonian description are no longer dynamical variables in Lg but merely abbreviations for combinations of the Lagrangian dynamical variables, which are here given by Eqs. (182). The correlation of the momenta πI ,πI of the base ﬁelds φI , φI to their derivatives are derived from the system Hamiltonian H via

∂φI ∂H µ = µ + igaIJµφJ + MIJ bJµ ∂x ∂πI

∂φI ∂H µ = µ − ig φJ aJIµ + bJµMIJ , (187) ∂x ∂πI

which represents the “minimal coupling rule” for our particular system. Thus, for any globally gauge-invariant Hamiltonian H(φI ,ππI , x), the amended La- grangian (186) with Eqs. (187) describes in the Lagrangian formalism the associated physical system that is invariant under local gauge transformations.

6.4. Klein-Gordon system Hamiltonian As an example, we consider the generalized Klein-Gordon Hamiltonian (Struckmeier and Reichau 2012) that describes an N-tuple of massless spin-0 ﬁelds

α HKG = πI πIα.

This Hamiltonian is clearly invariant under the inhomogeneous global gauge transformation (161). The reason for deﬁning a massless system Hamiltonian H is that a

mass term of the form φI MJI MJK φK that is contained in the general Klein-Gordon Hamiltonian is not invariant under the inhomogeneous gauge transformation from Eq. (161). According to Eqs. (186) and (187), the corresponding locally gauge- invariant Lagrangian L3,KG is then

α 1 αβ 1 αβ L3,KG = πI πIα − 4 fJK fKJαβ − 2 qJ qJαβ, (188) December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

86 J. Struckmeier, A. Redelbach

with ∂a ∂a f = KJν − KJµ + ig (a a − a a ) KJµν ∂xµ ∂xν KIν IJµ KIµ IJν ∂b ∂b q = Jν − Jµ + ig M˜ a M b − a M b + ig M˜ f φ Jµν ∂xµ ∂xν JI IKν KL Lµ IKµ KL Lν JI IKµν K ∂b ∂b q = Jν − Jµ − ig b M a − b M a M˜ + ig φ f M˜ Jµν ∂xµ ∂xν Lµ LK KIν Lν KL KIµ IJ I IKµν KJ ∂φI π = − iga φ − M b Iµ ∂xµ IJµ J IJ Jµ ∂φ π = I + ig φ a − b M . Iµ ∂xµ J JIµ Jµ IJ In a more explicit form, the gauge-invariant Lagrangian (188) thus writes

∂φ α ∂φ L = I + ig φ aα − b M I − iga φ − M b 3,KG ∂x J JI J IJ ∂xα IKα K IK Kα α 1 αβ 1 αβ − 4 fJK fKJαβ − 2 qI qIαβ. The terms in parentheses can be regarded as the “minimum coupling rule” for the actual system. With regard to the transformation prescription of Eq. (170), the corresponding product is obviously form-invariant under the inhomogeneous gauge transformation. Moreover, the Lagrangian contains a term that is proportional to the square of the 4-vector gauge fields bJ . With an orthogonal mass matrix M, this term simplifies to α 2 α bJ MIJ MIKbKα = (det M) bI bIα, which represents a Proca mass term for an N-tuple of possibly charged bosons with equal masses of det M. For the case N = 1, hence for a single base field φ, we may easily verify that the following twofold amended Klein-Gordon Lagrangian L3,KG

∂φ α ∂φ L = + ig φaα − m b − iga φ − mb − 1 f αβ f − 1 qαβq 3,KG ∂x ∂xα α α 4 αβ 2 αβ α is form-invariant under the combined local gauge transformation 1 ∂Λ φ 7→ Φ= φeiΛ + ϕ, a 7→ A = a + µ µ µ g ∂xµ ig 1 ∂Λ 1 ∂ϕ b 7→ B = b eiΛ − a + ϕ + . µ µ µ m µ g ∂xµ m ∂xµ The ﬁeld tensors then simplify to ∂a ∂a f = ν − µ µν ∂xµ ∂xν ∂b ∂b ig ∂a ∂a q = ν − µ + ig (a b − a b )+ ν − µ φ µν ∂xµ ∂xν ν µ µ ν m ∂xµ ∂xν ∂b ∂b ig ∂a ∂a q = ν − µ − ig b a − b a − φ ν − µ . µν ∂xµ ∂xν µ ν ν µ m ∂xµ ∂xν December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 87

2 α With m b bα, this locally gauge-invariant Lagrangian contains a mass term for the complex bosonic 4-vector ﬁeld b(x).

6.5. Dirac system Lagrangian A Dirac Lagrangian (Struckmeier and Reichau 2012)—describing N massless spin- 1 2 fields—that can regularly be Legendre-transformed into a corresponding Dirac Hamiltonian is given by i ∂ψ ∂ψ ∂ψ σαβ ∂ψ i L = ψ γα I − I γαψ + I I , σµν = (γµγν − γν γµ) . D 2 I ∂xα ∂xα I ∂xα i det M ∂xβ 2 Herein, det M stands for a coupling constant of dimension L−1 in order to ensure that the Hamiltonian takes on the correct dimension of L−4 as the spinor fields −3/2 ψI have the natural dimension [ψI ] = L . Due to the skew-symmetry of the σµν , the respective term does not contribute to the Euler-Lagrange equations (2). Thus, LD yields the correct Dirac equations for our given system of an N-tuple of uncoupled massless spinor fields ψI . Prior to being eligible to be converted into a locally gauge-invariant Lagrangian, the Lagrangian LD must be rendered globally gauge-invariant under the inhomogeneous transformation (161). This means that LD must be amended by terms that correspond to Eq. (163) with aµ ≡ 0 as only the inhomogeneous part of the transformation spoils the global gauge invariance of LD. The globally gauge-invariant Lagrangian L1,D is then i ∂ψ ∂ψ ∂ψ σαβ ∂ψ L = ψ γα I − M b − I − b M γαψ + I I . 1,D 2 I ∂xα IK Kα ∂xα Jα IJ I ∂xα i det M ∂xβ (189)

To obtain the corresponding locally gauge-invariant Lagrangian L3,D, we we follow the usual recipe to replace the partial derivatives by “covariant derivatives,” which in the actual case of an inhomogeneous gauge transformation is given by the “extended minimum coupling rule” from Eq. (170), and to finally add the “free field” terms for the gauge fields i ∂ψ i ∂ψ L = ψ γα I − iga ψ − 2M b − I + ig ψ a − 2b M γαψ 3,D 2 I ∂xα IKα K IK Kα 2 ∂xα J JIα Jα IJ I ∂ψ σαβ ∂ψ + I + ig ψ a − b M I − iga ψ − M b ∂xα J JIα Jα IJ i det M ∂xβ IKβ K IK Kβ det M − 1 f αβ f − q αβq . (190) 4 IJ JIαβ 2 J Jαβ

As the gauge fields bI always must have the same dimension as the base fields ψI , −3/2 the natural dimensions are [bI ] = L . In contrast to the 4-vector gauge fields aJK , which always describe bosonic particles, the gauge fields bI now have fermionic character. This accounts for the additional factor det M in front of the last term. December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

88 J. Struckmeier, A. Redelbach

Clearly, the form-invariance of L3,D is not affected by this constant factor. As the −3/2 result, qJµν with [qJ ]= L is given by ∂b ∂b (det M) q = Jν − Jµ +ig M˜ (a M b − a M b + f ψ ) . Jµν ∂xµ ∂xν JI IKν KL Lµ IKµ KL Lν IKµν K As in the case previous example, the Lagrangian contains a term that is proportional to the square of the 4-vector gauge fields bJ . With orthogonal M, this term simplifies to σαβ det M b M M b = b γαγβ − γβγα b . Jα IJ IK i det M Kβ 2 Jα Jβ This establishes a mass term in the gauge-invariant Lagrangian L 3,D. From the Lagrangian (190), the Euler-Lagrange equations for the base fields

ψI , ψI now follow as αβ ∂ ∂L3,D i α ∂ψI σ ∂aIKβ ∂ψK 1 ∂bKβ = − γ − g ψK + aIKβ + MIK ∂xα ∂ ∂ ψ 2 ∂xα det M ∂xα ∂xα ig ∂xα α I ∂L3,D i α ∂ψI α α = γ α + gγ aIKαψK − iMIKγ bKα ∂ψI 2 ∂x σαβ ∂ψ + g a J − iga ψ − M b det M IJα ∂xβ JKβ K JK Kβ and ∂ ∂L i ∂ψ ∂a ∂ψ ∂b σαβ 3,D = I γα + g ψ KIα + g K a + Kα i M ∂xβ ∂(∂ ψ ) 2 ∂xα K ∂xβ ∂xβ KIα ∂xβ IK det M β I ∂L3,D i ∂ψI α α α = − α γ + g ψK γ aKIα + i bKαγ MIK ∂ψI 2 ∂x ∂ψ σαβ − g J + ig ψ a − b M a . ∂xα K KJα Kα JK det M JIβ The second derivative terms in the base ﬁelds drop out due to the skew-symmetry of σαβ. The Euler-Lagrange equations thus simplify to ∂ψ iγα I + ga γαψ − i γαM b − 1 i σαβM q =0 ∂xα IKα K IK Kα 2 IK Kαβ ∂ψ i I γα − g ψ γαa − i b M γα − 1 i σαβ q M =0. ∂xα K KIα Kα IK 2 Kαβ KI We may convince ourselves by direct calculation that the ﬁeld equations for the

base ﬁelds ψI , ψI are form-invariant under the combined transformation deﬁned by Eqs. (161) and (166) ∂Ψ iγα + g A γαΨ − iγαMBB − 1 i σαβ MQQ =0 ∂xα α α 2 αβ ∂ψψ 1 ∂U U iγα +gaa γαψ − iγαMbb − 1 iσαβMqq +g Uaa U † + U † − A γαϕ =0 ∂xα α α 2 αβ α ig ∂xα α =0 ∂ψψ ⇒ iγα + gaa γαψ − iγαMbb − 1 iσαβMqq =0|. {z } ∂xα α α 2 αβ December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 89

In particular, the terms proportional to γα as well as the term proportional to σαβ are separately form-invariant. For the case of a system with a single spinor ψ, the locally gauge-invariant Dirac equation reduces to ∂ψ iγα + ga γαψ − imγαb ∂xα α α ig ∂a ∂a ∂b ∂b − 1 γβγα β − α ψ + β − α + ig (a b − a b ) =0. 2 m ∂xα ∂xβ ∂xα ∂xβ β α α β The mass m thus acts as the second coupling constant. The sum of terms linear in the γµ as well as the sum of terms quadratic in the γµ are separately invariant under the combined transformation of base ﬁelds ψ and gauge ﬁelds aµ,bµ 1 ∂Λ a 7→ A = a + , ψ 7→ Ψ= ψeiΛ + ϕ µ µ µ g ∂xµ ig 1 ∂Λ 1 ∂ϕ b 7→ B = b eiΛ − a + ϕ + . µ µ µ m µ g ∂xµ m ∂xµ

6.6. Canonical transformations of the Dirac Hamiltonian 6.6.1. Shift of the canonical momentum vectors

The canonical momenta emerging from the Dirac Lagrangian LD (Eq. (90)) are ∂L i ∂L i πµ = D = ψγµ, πµ = D = − γµψ, ∂(∂µψ) 2 ∂ ∂µψ 2

′ whereas those emerging from LD (Eq. (93)) were already derived in Eq. (95),

µ i i ∂ψ i i ∂ψ Π = ψγµ − σαµ, Πµ = − γµψ − σµα . 2 m ∂xα 2 m ∂xα We may regard this as a transformation of the canonical momenta,

µ i ∂ψ i ∂ψ Π = πµ − σαµ, Πµ = πµ − σµα , (191) m ∂xα m ∂xα which is uniquely determined by an explicitly xν -dependent generating function of µ µ µ type F2 (ψ, ψ, Π , Π , x),

µ i ∂ψ ∂ψ F µ = ψΠµ + Π ψ + ψσµα + σαµψ . 2 m ∂xα ∂xα As the derivatives of ψ and ψ are no canonical variables, these quantities must be treated as explicitly x-dependent coeﬃcients. According to the general rules (19), the transformation of the momenta is given by Eq. (191),

µ µ ∂F µ i ∂ψ ∂F i ∂ψ πµ = 2 = Π + σαµ, πµ = 2 = Πµ + σµα . ∂ψ m ∂xα ∂ψ m ∂xα December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

90 J. Struckmeier, A. Redelbach

The corresponding rules for the ﬁelds ψ and ψ yield identities, µ µ µ ∂F2 µ µ ∂F2 µ Ψδν = = ψδν , Ψδν = ν = ψδν . ∂Πν ∂Π The transformation rule for the Hamiltonian follows from the explicit x-dependence of the generating function

β 2 2 ′ ∂F2 i βα ∂ ψ ∂ ψ αβ HD − HD = = ψσ + σ ψ =0. ∂xβ m ∂xα∂xβ ∂xβ∂xα expl

Again, both terms in parentheses vanish as they involve summations over purely symmetric and skew-symmetric factors in the α, β. For the same reason, the divergences of the original and the transformed vectors of canonical momenta coincide,

β ∂Π ∂πβ i ∂2ψ ∂πβ = − σαβ = ∂xβ ∂xβ m ∂xβ∂xα ∂xβ ∂Πβ ∂πβ i ∂2ψ ∂πβ = − σαβ = . ∂xβ ∂xβ m ∂xβ∂xα ∂xβ The primed and the unprimed sets of canonical momenta are thus equivalent as only their divergences are determined by the Hamiltonian H. The transformation of the ′ Dirac Lagrangian LD to the equivalent Lagrangian LD thus appears in the Hamil- µ tonian formalism as a shift of the canonical momentum vectors πµ → Πµ, πµ → Π that maintains their divergences, hence that emerge from the same Hamiltonian HD.

6.6.2. Interchange of the canonical variables The generating function of a canonical transformation that interchanges ﬁelds and momenta is

µ 1 µ 1 µ F1 ψ, Ψ, ψ, Ψ = − 2 ψγ Ψ+ 2 Ψγ ψ. (192) q q The general transformation rules (16) yield for the particular generating function (192) ∂F µ ∂F µ πµ = 1 = − 1 γµΨ, Πµ = − 1 = − 1 γµψ, H′ = H ∂ψ 2 ∂Ψ 2 µ q µ q ∂F µ ∂F πµ = 1 = 1 Ψγµ, Π = − 1 = 1 ψγµ. (193) ∂ψ 2 ∂Ψ 2 q q The Dirac Hamiltonian was derived in Sect. 4.7, 1 1 2 H = im ψγ πα − πατ πβ − παγ ψ + m ψψ. (194) D 6 α αβ 6 α 3 As the generating function (192) does not explicitly depend on the independent variables, xµ, we have H′ = H. The transformed Dirac Hamiltonian is thus obtained December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 91

by expressing the original canonical variables in terms of the new ones. Explicitly, the four components of the Hamiltonian (194) transform as

i i 4 2 −i πατ πβ = Ψγατ γβΨ= Ψ 1 Ψ= ΨΨ αβ 2 αβ 2 3i 3 2 2 3i α ψψ = ψγατ γβψ = i 1 ψγα τ 1 γβψ = −i Π τ Πβ 3 3 4 αβ 2 αβ 2 αβ q q α 1 α 1 α α π γαψ = 2 Ψγ γαψ = Ψγα 2 γ ψ = −ΨγαΠ

α q 1 α q1 α α ψγαπ = −ψγα 2 γ Ψ= − 2 ψγ γαΨ= −Π γαΨ.

q q ′ The complete transformed Dirac Hamiltonian HD is now given by

1 α 1 α 2 H′ = im Ψγ Πα − Π τ Πβ − Π γ Ψ + m ΨΨ, D 6 α αβ 6 α 3 and obviously has exactly the form of the original one. We conclude that it is precisely the Dirac Hamiltonian density (194) that has the additional symmetry to be invariant under the canonical transformation that interchanges the expressions γµψ with the momenta πµ.

6.6.3. Local U(1) gauge theory applied to the Dirac Hamiltonian

With ψ(x) denoting a four-component Dirac spinor, the Dirac Hamiltonian HD from 1 Eq. (98) describes a single spin- 2 ﬁeld. For the local gauge transformation (131) of ψ(x), this means that the particle indices are restricted to the case I,J,K,L = 1 in ∗ −1 the general transformation rule (137) of gauge ﬁelds aµ(x). As u(x),u (x)= u (x) thus denote complex numbers rather than matrices in that case, the transformation rules for ψ, πµ and their adjoints simplify to

Ψ= u(x)ψ, Ψ= u∗(x)ψ µ Πµ = u(x)πµ, Π = u∗(x)πµ.

Due to the gauge ﬁeld matrix being Hermitian, aJI = aIJ , the gauge ﬁeld components a11µ(x) ≡ aµ(x) are then real numbers obeying the transformation rules 1 ∂ A (x)= a (x)+ ln u(x). µ µ iq ∂xµ We may express u(x) in terms of a phase function θ(x)

u(x)= eiθ(x), u∗(x)= e−iθ(x).

The transformation rule (137) for the gauge ﬁelds then simpliﬁes to 1 ∂θ(x) A = a + . µ µ q ∂xµ December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

92 J. Struckmeier, A. Redelbach

According to Eq. (146), the supplemented Dirac Hamiltonian H¯D = HD +Hg, which is invariant under the local phase transformation Ψ = ψ exp(iθ(x)), is then given by 1 1 2 H¯ = −im πατ πβ + παγ ψ − ψγ πα + m ψψ D αβ 6 α 6 α 3 1 1 ∂a ∂a + iq παψ − ψπα a − pαβp + pαβ α + β . (195) α 4 αβ 2 ∂xβ ∂xα 1 This Hamiltonian describes both the dynamics of a spin- 2 particle field ψ(x) with mass m and a massless 4-vector field a(x) in conjunction with their mutual interaction. The coupling strength of both quantities is governed by the coupling constant q. The relation of the canonical momenta pµν to the derivatives of the fields, ν ∂aµ/∂x , is obtained from the first canonical field equation (5). This means for the Dirac Hamiltonian HD, and equally for the gauge-invariant Hamiltonian H¯D, ∂a ∂H¯ 1 1 ∂a ∂a µ = D = − p + µ + ν , ∂xν ∂pµν 2 µν 2 ∂xν ∂xµ hence ∂a ∂a p = ν − µ . µν ∂xµ ∂xν

We observe that pµν happens to be skew-symmetric in the indices µ,ν. Here, this feature emerges directly from the canonical gauge theory presented in Sect. 5.8 and does not need to be postulated. On the other hand, with a skew-symmetric pµν it follows that pµν is skew-symmetric as well. Therefore, the value of the last term in the Hamiltonian (195) vanishes as the sum in parentheses is symmetric in α, β. As this term only contributes to the first canonical equation, we may omit this term from the Hamiltonian (195) if we simultaneously define pµν to be skew-symmetric in µ,ν. The Hamiltonian H¯D is then equivalent to 1 1 2 H¯ = −im πατ πβ + παγ ψ − ψγ πα + m ψψ D αβ 6 α 6 α 3 1 + iq παψ − ψπα a − pαβp , p Def= −p . (196) α 4 αβ µν νµ References 1. W. Greiner, B. Müller, and J. Rafelski, Quantum Electrodynamics of Strong Fields, (Springer-Verlag, Berlin, 1985). 2. S. Weinberg, The Quantum Theory of Fields, Vol. I, (Cambridge University Press, Cambridge, 1996). 3. L. H. Ryder, Quantum Field Theory, (Cambridge University Press, Cambridge, 2006). 4. D. Griffiths, Introduction to Elementary Particles, (J. Wiley & Sons, New York, 1987). 5. J. V. José, E. J. Saletan, Classical Dynamics, (Cambridge University Press, Cam- bridge, 1998). December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte

Covariant Hamiltonian Field Theory 93

6. Th. De Donder, Théorie Invariantive Du Calcul des Variations, (Gaulthier-Villars & Cie., Paris, 1930). 7. H. Weyl, Geodesic Fields in the Calculus of Variation for Multiple Integrals, in Annals of Mathematics 36 607 (1935). 8. I. V. Kanatchikov, Canonical Structure of Classical Field Theory in the Polymomen- tum Phase Space, in Reports on Mathematical Physics 41 49 (1998). 9. M. J. Gotay, J. Isenberg, J. E. Marsden, and R. Montgomery, Momentum Maps and Classical Fields, Part I, Covariant Field Theory, arXiv:physics/9801019 v2, 19 Aug 2004. 10. H. A. Kastrup, Canonical Theories of Lagrangian Dynamical Systems in Physics, in Physics Reports 101 1 (1983). 11. P. C. Paufler, Multisymplektische Feldtheorie, Thesis, University of Freiburg im Breis- gau, Germany (2001). 12. M. Forger, C. Paufler, and H. Römer, The Poisson Bracket for Poisson Forms in Multisymplectic Field Theory, in Reviews in Mathematical Physics 15 705 (2003). 13. A. Echeverr´ıa-Enr´ıquez, M. C. Muñoz-Lecanda, and N. Román-Roy, Geometry of Lagrangian First-oder Classical Field Theories, in Fortschr. Phys. 44 235 (1996). 14. G. Sardanashvily, Generalized Hamiltonian Formalism for Field Theory, (World Sci- entific Publishing Co., Singapore, 1995). 15. V. V. Kisil, p-mechanics as a physical theory: an introduction, in J. Phys. A: Math. Gen. 37 183 (2004). 16. D. J. Saunders, The Geometry of Jet Bundles, (Cambridge University Press, Cam- brigde, 1989). 17. Ch. Günther, The Polysymplectic Hamiltonian Formalism in Field Theory and Cal- culus of Variations I: The Local Case, in J. Differential Geometry 25 23 (1987). 18. W. Greiner, Classical Mechanics, (Springer Verlag, New York, 2003). 19. H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed. (Pearson, Addison- Wesley, Upper Saddle River, NJ, 2002). 20. D. Muˇsicki, On canonical formalism in field theory with derivatives of higher order— canonical transformations, in J. Phys. A: Math. Gen. 11 39 (1978). 21. S. Gasiorowicz, Elementary particle physics, (Wiley, New York, 1966). 22. J. von Rieth, The Hamilton-Jacobi theory of De Donder and Weyl applied to some relativistic field theories, in J. Math. Phys. 25 1102 (1984). 23. Ta-Pei Cheng and Ling-Fong Li, Gauge theory of elementary particle physics, (Oxford University Press, Oxford, 2000). 24. J. Struckmeier, Hamiltonian dynamics on the symplectic extended phase space for autonomous and non-autonomous systems, J. Phys. A: Math. Gen. 38 1257 (2005). 25. J. Struckmeier, Extended Hamilton-Lagrange formalism and its application to Feyn- man’s path integral for relativistic quantum physics, Int. J. Mod. Phys. 18 79 (2009). 26. H. Stephani, Relativity, (Cambridge University Press, Cambridge, 2004). 27. L. H. Ryder, Introduction to General Relativity, (Cambridge University Press, Cam- bridge, 2009).