Gauge invariant theories of the generalized chiral Schwinger model

The gauge invariant theories of the generalized chiral Schwinger model are constructed in terms of two schemes with and without Wess-Zumino terms, respectively. Following the former scheme, we calculate the Wess-Zumino term which cancels the gauge anomaly, and then constitute the gauge invariant theory by adding the Wess-Zumino term to the original Lagrangian of the model. According to the latter, we modify the original Hamiltonian by adding a term composed of constraints of the model. It is so designed that the theory described by the modified Hamiltonian and its corresponding first-order Lagrangian maintains gauge invariance. We show by the canonical Dirac method that each of the two gauge invariant theories has the same physical spectrum as that of the original gauge noninvariant formulation.     

Gauge invariant theories of the generalized chiral Schwinger model

The gauge invariant theories of the generalized chiral Schwinger model are constructed in terms of two schemes with and without Wess-Zumino terms, respectively. Following the former scheme, we calculate the Wess-Zumino term which cancels the gauge anomaly, and then constitute the gauge invariant theory by adding the Wess-Zumino term to the original Lagrangian of the model. According to the latter, we modify the original Hamiltonian by adding a term composed of constraints of the model. It is so designed that the theory described by the modified Hamiltonian and its corresponding first-order Lagrangian maintains gauge invariance. We show by the canonical Dirac method that each of the two gauge invariant theories has the same physical spectrum as that of the original gauge noninvariant formulation.     

The Limit of Noether Conserved Charges is the Number of Primary First-Class Constraints in a Constrained System

We discuss the stationarity of generator G for gauge symmetries in two directions.One is to the motion equations defined by total Hamiltonian H_T,and gives that the number of the independent coefficients in the generator G is not greater than the number of the primary first-class constraints,and the number of Noether conserved charges is not greater than that of the primary first-class constraints,too.The other is to the variances of canonical variables deduced from the generator G,and gives the variances of Lagrangian multipliers contained in extended Hamiltonian H_E.And a second-class constraint generated by a first-class constraint may imply a new first-class constraint which can be combined by introducing other second-class constraints.Finally,we supply two examples.One with three first-class constraints (two is primary and one is secondary) has two Noether conserved charges,and the secondary first-class constraint is combined by three second-class constraints which are a secondary and two primary second-class constraints.The other with two first-class constraints (one is primary and one is secondary) has one Noehter conserved charge.     

A Gauge Invariant Description for the General Conic Constrained Particle from the FJBW Iteration Algorithm

We consider a second-degree algebraic curve describing a general conic constraint imposed on the motion of a massive spinless particle. The problem is trivial at classical level but becomes involved and interesting concerning its quantum counterpart with subtleties in its symplectic structure and symmetries. We start with a second-class version of the general conic constrained particle, which encompasses previous versions of circular and elliptical paths discussed in the literature. By applying the symplectic FJBW iteration program, we proceed on to show how a gauge invariant version for the model can be achieved from the originally second-class system. We pursue the complete constraint analysis in phase space and perform the Faddeev-Jackiw symplectic quantization following the Barcelos-Wotzasek iteration program to unravel the essential aspects of the constraint structure. While in the standard Dirac-Bergmann approach there are four second-class constraints, in the FJBW they reduce to two. By using the symplectic potential obtained in the last step of the FJBW iteration process, we construct a gauge invariant model exhibiting explicitly its BRST symmetry. We obtain the quantum BRST charge and write the Green functions generator for the gauge invariant version. Our results reproduce and neatly generalize the known BRST symmetry of the rigid rotor, clearly showing that this last one constitutes a particular case of a broader class of theories.     

Faddeev–Jackiw quantization of the higher derivative Chern–Simons gauge theory in the first-order formalism

We analyse the physical constraints of the higher derivative Chern–Simons gauge model by means of Faddeev–Jackiw symplectic approach in the first-order formalism. Within such framework, we systematically determine the zero-mode structure of the corresponding symplectic matrix. In addition, we calculate the Faddeev–Jackiw quantum brackets by choosing appropriate gauge-fixing conditions and evaluate the determinant of the non-singular symplectic matrix as well as the transition-amplitude. Finally, we present a detailed Hamiltonian analysis using Dirac–Bergmann algorithm method and show that the Dirac brackets coincide with the FJ brackets when all the second-class constraints are treated as zero equations.     

Dirac Quantization of Noncommutative Abelian Proca field

Dirac formalism of Hamiltonian constraint systems is studied for the noncommutative Abelian Proca field. It is shown that the system of constraints are of second class in agreement with the fact that the Proca field is not gauge invariant. Then, the system of second class constraints is quantized by introducing Dirac brackets in the reduced phase space.     

相关于弦论的规范不变自对偶场的Faddeev-Jackiw量子化

用Faddeev-Jackiw(FJ)方法对与规范场偶合的规范自对偶场进行了研究, 获得了一个新的辛Lagrangian密度, 导出了此系统的FJ广义括号, 并对其进行了FJ量子化. 进而把FJ方法和Dirac方法进行了比较, 发现在对此系统的量子化中, 两种方法所给出的量子化结果完全是等价的. 通过分析可知FJ方法比Dirac方法要简单, 因FJ方法不需要区分初级约束与次级约束, 而且也不需要区分第一类约束和第二类约束. 故与Dirac方法相比, FJ方法是一种计算上更为经济和有效的量子化方法.     

Gauge Symmetries and Dirac Conjecture

The gauge symmetries of a constrained system can be deduced from the gauge identities with Lagrange method, or the first-class constraints with Hamilton approach. If Dirac conjecture is valid to a dynamic system, in which all the first-class constraints are the generators of the gauge transformations, the gauge transformations deduced from the gauge identities are consistent with these given by the first-class constraints. Once the equivalence vanishes to a constrained system, in which Dirac conjecture would be invalid. By using the equivalence, two counterexamples and one example to Dirac conjecture are discussed to obtain defined results.     

Equivalent Theories and Changing Hamiltonian Observables in General Relativity

Change and local spatial variation are missing in Hamiltonian general relativity according to the most common definition of observables as having 0 Poisson bracket with all first-class constraints. But other definitions of observables have been proposed. In pursuit of Hamiltonian–Lagrangian equivalence, Pons, Salisbury and Sundermeyer use the Anderson–Bergmann–Castellani gauge generator G, a tuned sum of first-class constraints. Kucha? waived the 0 Poisson bracket condition for the Hamiltonian constraint to achieve changing observables. A systematic combination of the two reforms might use the gauge generator but permit non-zero Lie derivative Poisson brackets for the external gauge symmetry of General Relativity. Fortunately one can test definitions of observables by calculation using two formulations of a theory, one without gauge freedom and one with gauge freedom. The formulations, being empirically equivalent, must have equivalent observables. For de Broglie-Proca non-gauge massive electromagnetism, all constraints are second-class, so everything is observable. Demanding equivalent observables from gauge Stueckelberg–Utiyama electromagnetism, one finds that the usual definition fails while the Pons–Salisbury–Sundermeyer definition with G succeeds. This definition does not readily yield change in GR, however. Should GR’s external gauge freedom of general relativity share with internal gauge symmetries the 0 Poisson bracket (invariance), or is covariance (a transformation rule) sufficient? A graviton mass breaks the gauge symmetry (general covariance), but it can be restored by parametrization with clock fields. By requiring equivalent observables, one can test whether observables should have 0 or the Lie derivative as the Poisson bracket with the gauge generator G. The latter definition is vindicated by calculation. While this conclusion has been reported previously, here the calculation is given in some detail.     

Gauge identities and the Dirac conjecture

The gauge symmetries of a general dynamical system can be systematically obtained following either a Hamiltonian or a Lagrangean approach. In the former case, these symmetries are generated, according to Dirac's conjecture, by the first class constraints. In the latter approach such local symmetries are reflected in the existence of so called gauge identities. The connection between the two becomes apparent, if one works with a first order Lagrangean formulation. Our analysis applies to purely first class systems. We show that Dirac's conjecture applies to first class constraints which are generated in a particular iterative way, regardless of the possible existence of bifurcations or multiple zeroes of these constraints. We illustrate these statements in terms of several examples.     

Obtaining gauge invariant actions via symplectic embedding formalism

The concept of gauge invariance is one of the most subtle and useful concepts in modern theoretical physics. It is one of the Standard Model cornerstones. The main benefit due to the gauge invariance is that it can permit the comprehension of difficult systems in physics with an arbitrary choice of a reference frame at every instant of time. It is the objective of this work to show a path of obtaining gauge invariant theories from non‐invariant ones. Both are named also as first‐ and second‐class theories respectively, obeying Dirac's formalism. Namely, it is very important to understand why it is always desirable to have a bridge between gauge invariant and non‐invariant theories. Once established, this kind of mapping between first‐class (gauge invariant) and second‐class systems, in Dirac's formalism can be considered as a sort of equivalence. This work describe this kind of equivalence obtaining a gauge invariant theory starting with a non‐invariant one using the symplectic embedding formalism developed by some of us some years back. To illustrate the procedure it was analyzed both Abelian and non‐Abelian theories. It was demonstrated that this method is more convenient than others. For example, it was shown exactly that this embedding method used here does not require any special modification to handle with non‐Abelian systems.     

Canonical and D-transformations in theories with constraints

We describe a class of transformations in a super phase space (we call them D-transformations) which play the role of ordinary canonical transformations in theories with second-class constraints. Namely, in such theories they preserve the form invariance of equations of motion, their quantum analogs are unitary transformations, and the measure of integration in the corresponding Hamiltonian path integral is invariant under these transformations.     

Gauge invariance,anomalies, and the chiral Schwinger model

After having justified the gauge invariant version of the chiral Schwinger model we perform canonical quantization via Dirac brackets. The constraints are first class, exhibiting gauge invariance. As a result we find that this is the reason for the consistency of the model of Jackiw and Rajaraman.     

Review: Dirac's Observables for the Rest-Frame Instant Form of Tetrad Gravity in a Completely Fixed 3-Orthogonal Gauge

We define the rest-frame instant form of tetrad gravity restricted to Christodoulou-Klainermann spacetimes. After a study of the Hamiltonian group of gauge transformations generated by the 14 first class constraints of the theory, we define and solve the multitemporal equations associated with the rotation and space diffeomorphism constraints, finding how the cotriads and their momenta depend on the corresponding gauge variables. This allows to find a quasi-Shanmugadhasan canonical transformation to the class of 3-orthogonal gauges and to find the Dirac observables for superspace in these gauges. The construction of the explicit form of the transformation and of the solution of the rotation and supermomentum constraints is reduced to solve a system of elliptic linear and quasi-linear partial differential equations. We then show that the superhamiltonian constraint becomes the Lichnerowicz equation for the conformal factor of the 3-metric and that the last gauge variable is the momentum conjugated to the conformal factor. The gauge transformations generated by the superhamiltonian constraint perform the transitions among the allowed foliations of spacetime, so that the theory is independent from its 3+1 splittings. In the special 3-orthogonal gauge defined by the vanishing of the conformal factor momentum we determine the final Dirac observables for the gravitational field even if we are not able to solve the Lichnerowicz equation. The final Hamiltonian is the weak ADM energy restricted to this completely fixed gauge.     

Gauge invariance and the dirac equation

The gauge invariance of the Dirac equation is reviewed and gauge-invariant operators are defined. The Hamiltonian is shown to be gauge dependent, and an energy operator is defined which is gauge invariant. Gauge-invariant operators corresponding to observables are shown to satisfy generalized Ehrenfest theorems. The time rate of change of the expectation value of the energy operator is equal to the expectation value of the power operator. The virial theorem is proved for a relativistic electron in a time-varying electromagnetic field. The conventional approach to probability amplitudes, using the eigenstates of the unperturbed Hamiltonian, is shown in general to be gauge dependent. A gaugeinvariant procedure for probability amplitudes is given, in which eigenstates of the energy operator are used. The two methods are compared by applying them to an electron in a zero electromagnetic field in an arbitrary gauge. Presented at the Dirac Symposium, Loyola University, New Orleans, May 1981.     

Quantenphysikalischer Ursprung der Eichidee

The Quantum Physical Origin of the Gauge Idea To consider quantum physics as an interplay of creation and annihilation processes has the consequence that gauge field theories are not only possible but necessary. Since the complex conjugate phase factors of each pair of fermion creators and annihilators can be arbitrary chosen, quantum field theories must be completely phase invariant. Unfortunately, even globally the Dirac equation for systems of free fermions is not phase invariant. The Dirac matrices are namely transformed, if we multiply the spinor components by different constant phase factors. The Dirac equations before and after the transformation are however physically equivalent. We may therefore say: Systems of free fermions will be completely described, only if we consider the class of all equivalent Dirac equations. Since Dirac's commutation relations are unitarily invariant, the class equivalent Dirac equations is invariant under all transformations of the group U 4. Unitary diagonal matrices yield arbitrary phase transformations. Hence, gauge fields of the group U 4 are compatible with the postulate of general phase invariance. These gauge file are so similar to the QED that we may speak of an “extended quantum electrodynamics”, EQE. Here, we will show that EQE exists. The invariant subgroup U 1 U 4 yields QED. The complementary subgroup SU 4 includes four subgroups SU 3, there subgroups O 4, and six subgroups SU 2. The latter ones may yield three pairs of quarks and three pairs of leptons, where the quarks form a group SU 3. More than two times three pairs of elementary fermions does not exist in in EQE Probably, EQE is different from the United EQD and QCD. However, it should be a promising version of a field theory in elementary particle physics, because it follows from an existing symmetry of the empirically wel founded Dirac theory. EQE is therefore free from hypothesis in the Newtonian sense of the word. Whatever it will finally mean, it cannot be rejected, since phase invariance must be required. The invention of new symmetries and the acception of a bie number of independent spinor components is dispensable or must be postponed at least.     

Gauge invariance based on the extrinsic geometry in the rigid string

The string model with the extrinsic curvature is studied which is a gauge invariant field theory with higher order derivatives. We present an equivalent action without any higher order derivatives which keeps the gauge invariance. We point out the difficulty caused by the second class constraints in Dirac's canonical method. Following a new method for dynamical systems with second class constraints, we construct an equivalent model which has no second class constrants but as a new gauge invariance. This gauge invariance guarantees the equivalence between the original model and the new one. We show that the model can be quantized in this formalism. We find the unitarity violation of the model.     

Electromagnetic interaction in theory with Lorentz invariant CPT violation

An attempt is made to incorporate the electromagnetic interaction in a Lorentz invariant but CPT violating non-local model with particle–antiparticle mass splitting, which is regarded as a modified QED. The gauge invariance is maintained by the Schwinger non-integrable phase factor but the electromagnetic interaction breaks C, CP and CPT symmetries. Implications of the present CPT breaking scheme on the electromagnetic transitions and particle–antiparticle pair creation are discussed. The CPT violation such as the one suggested in this Letter may open a new path to the analysis of baryon asymmetry since some of the Sakharov constraints are expected to be modified.     

Hamiltonian Dynamics for Einstein’s Action in <Emphasis Type="Italic">G</Emphasis>→0 Limit

The Hamiltonian analysis for the Einstein’s action in G→0 limit is performed. Considering the original configuration space without involve the usual ADM variables we show that the version G→0 for Einstein’s action is devoid of physical degrees of freedom. In addition, we will identify the relevant symmetries of the theory such as the extended action, the extended Hamiltonian, the gauge transformations and the algebra of the constraints. As complement part of this work, we develop the covariant canonical formalism where will be constructed a closed and gauge invariant symplectic form. In particular, using the geometric form we will obtain by means of other way the same symmetries that we found using the Hamiltonian analysis.