It is a commonplace in the philosophy of physics that any local physical theory can be represented using arbitrary coordinates, simply by using tensor calculus. On the other hand, the physics literature often claims that spinors \emph{as such} cannot be represented in coordinates in a curved space-time. These commonplaces are inconsistent. What general covariance means for theories with fermions, such as electrons, is thus unclear. In fact both commonplaces are wrong. Though it is not widely known, Ogievetsky and Polubarinov constructed (...) spinors in coordinates in 1965, enhancing the unity of physics and helping to spawn particle physicists' concept of nonlinear group representations. Roughly and locally, these spinors resemble the orthonormal basis or "tetrad" formalism in the symmetric gauge, but they are conceptually self-sufficient and more economical. The typical tetrad formalism is de-Ockhamized, with six extra field components and six compensating gauge symmetries to cancel them out. The Ogievetsky-Polubarinov formalism, by contrast, is Ockhamized, with most of the fluff removed. As developed nonperturbatively by Bilyalov, it admits any coordinates at a point, but "time" must be listed first. Here "time" is defined in terms of an eigenvalue problem involving the metric components and the matrix $diag$, the product of which must have no negative eigenvalues in order to yield a real symmetric square root that is a function of the metric. Thus even formal general covariance requires reconsideration; the atlas of admissible coordinate charts should be sensitive to the types and \emph{values} of the fields involved. Apart from coordinate order and the usual spinorial two-valuedness, Ogievetsky-Polubarinov spinors form, with the metric, a nonlinear geometric object, for which important results on Lie and covariant differentiation are recalled. Such spinors avoid a spurious absolute object in the Anderson-Friedman analysis of substantive general covariance. They also permit the gauge-invariant localization of the infinite-component gravitational energy in General Relativity. Density-weighted spinors exploit the conformal invariance of the massless Dirac equation to show that the volume element is absent. Thus instead of an arbitrary nonsingular matrix with 16 components, 6 of which are gauged away by a new local $O$ gauge group and one of which is irrelevant due to conformal covariance, one can, and presumably should, use density-weighted Ogievetsky-Polubarinov spinors coupled to the 9-component symmetric square root of the part of the metric that fixes null cones. Thus $\frac{7}{16}$ of the orthonormal basis is eliminated as surplus structure. Greater unity between spinors and tensors and the like is achieved, such as regarding conservation laws. Regarding the conventionality of simultaneity, an unusually wide range of $\epsilon$ values is admissible, but some extreme values are inadmissible. Standard simultaneity uniquely makes the spinor transformation law linear and independent of the metric, because transformations among the standard Cartesian coordinate systems fall within the conformal group, for which the spinor transformation law is linear. The surprising mildness of the restrictions on coordinate order as applied to the Schwarzschild solution is exhibited. (shrink)

It is a commonplace in the foundations of physics, attributed to Kretschmann, that any local physical theory can be represented using arbitrary coordinates, simply by using tensor calculus. On the other hand, the physics and mathematics literature often claims that spinors \emph{as such} cannot be represented in coordinates in a curved space-time. These commonplaces are inconsistent. What general covariance means for theories with fermions is thus unclear. In fact both commonplaces are wrong. Though it is not widely known, Ogievetsky and (...) Polubarinov constructed spinors in coordinates in 1965, enhancing the unity of physics and helping to spawn particle physicists' concept of nonlinear group representations. Roughly and locally, OP spinors resemble the orthonormal basis or tetrad formalism in the symmetric gauge, but they are conceptually self-sufficient and more economical. The typical tetrad formalism is thus de-Ockhamized, with six extra field components and six compensating gauge symmetries to cancel them out. As developed nonperturbatively by Bilyalov, OP spinors admit any coordinates at a point, but `time' must be listed first; `time' is defined in terms of an eigenvalue problem involving the metric components and the matrix $diag$, the product of which must have no negative eigenvalues. Thus even formal general covariance requires reconsideration; the atlas of admissible coordinate charts should be sensitive to the types and \emph{values} of the fields involved. Apart from coordinate order and the usual spinorial two-valuedness, Ogievetsky-Polubarinov spinors form, with the metric, a nonlinear geometric object. Important results on Lie and covariant differentiation are recalled and applied. The rather mild consequences of the coordinate order restriction are explored in two examples: the question of the conventionality of simultaneity in Special Relativity, and the Schwarzschild solution in General Relativity. (shrink)

In General Relativity in Hamiltonian form, change has seemed to be missing, defined only asymptotically, or otherwise obscured at best, because the Hamiltonian is a sum of first-class constraints and a boundary term and thus supposedly generates gauge transformations. Attention to the gauge generator G of Rosenfeld, Anderson, Bergmann, Castellani et al., a specially _tuned sum_ of first-class constraints, facilitates seeing that a solitary first-class constraint in fact generates not a gauge transformation, but a bad physical change in electromagnetism or (...) General Relativity. The change spoils the Lagrangian constraints, Gauss's law or the Gauss-Codazzi relations describing embedding of space into space-time, in terms of the physically relevant velocities rather than auxiliary canonical momenta. But the resemblance between the gauge generator G and the Hamiltonian H leaves still unclear where objective change is in GR. Insistence on Hamiltonian-Lagrangian equivalence, a theme emphasized by Castellani, Sugano, Pons, Salisbury, Shepley and Sundermeyer among others, holds the key. Taking objective change to be ineliminable time dependence, one recalls that there is change in vacuum GR just in case there is no time-like vector field xi^a satisfying Killing's equation L_xi g_mn=0, because then there exists no coordinate system such that everything is independent of time. Throwing away the spatial dependence of GR for convenience, one finds explicitly that the time evolution from Hamilton's equations is real change just when there is no time-like Killing vector. The inclusion of a massive scalar field is simple. No obstruction is expected in including spatial dependence and coupling more general matter fields. Hence change is real and local even in the Hamiltonian formalism. The considerations here resolve the Earman-Maudlin standoff over change in Hamiltonian General Relativity: the Hamiltonian formalism is helpful, and, suitably reformed, it does not have absurd consequences for change and observables. Hence the classical problem of time is resolved. The Lagrangian-equivalent Hamiltonian analysis of change in General Relativity is compared to Belot and Earman's treatment. The more serious quantum problem of time, however, is not automatically resolved due to issues of quantum constraint imposition. (shrink)

In Dirac-Bergmann constrained dynamics, a first-class constraint typically does not _alone_ generate a gauge transformation. By direct calculation it is found that each first-class constraint in Maxwell's theory generates a change in the electric field E by an arbitrary gradient, spoiling Gauss's law. The secondary first-class constraint p^i,_i=0 still holds, but being a function of derivatives of momenta, it is not directly about E. Only a special combination of the two first-class constraints, the Anderson-Bergmann -Castellani gauge generator G, leaves E (...) unchanged. This problem is avoided if one uses a first-class constraint as the generator of a _canonical transformation_; but that partly strips the canonical coordinates of physical meaning as electromagnetic potentials and makes the electric field depend on the smearing function, bad behavior illustrating the wisdom of the Anderson-Bergmann Lagrangian orientation of interesting canonical transformations. The need to keep gauge-invariant the relation dot{q}- dH/dp= -E_i -p^i=0 supports using the primary Hamiltonian rather than the extended Hamiltonian. The results extend the Lagrangian-oriented reforms of Castellani, Sugano, Pons, Salisbury, Shepley, _etc._ by showing the inequivalence of the extended Hamiltonian to the primary Hamiltonian even for _observables_, properly construed in the sense implying empirical equivalence. Dirac and others have noticed the arbitrary velocities multiplying the primary constraints outside the canonical Hamiltonian while apparently overlooking the corresponding arbitrary coordinates multiplying the secondary constraints _inside_ the canonical Hamiltonian, and so wrongly ascribed the gauge quality to the primaries alone, not the primary-secondary team G. Hence the Dirac conjecture about secondary first-class constraints rests upon a false presupposition. The usual concept of Dirac observables should also be modified to employ the gauge generator G, not the first-class constraints separately, so that the Hamiltonian observables become equivalent to the Lagrangian ones such as the electromagnetic field F. (shrink)

An Introduction to Quantum Field Theory is a textbook intended for the graduate physics course covering relativistic quantum mechanics, quantum electrodynamics, and Feynman diagrams. The authors make these subjects accessible through carefully worked examples illustrating the technical aspects of the subject, and intuitive explanations of what is going on behind the mathematics. After presenting the basics of quantum electrodynamics, the authors discuss the theory of renormalization and its relation to statistical mechanics, and introduce the renormalization group. This discussion sets the (...) stage for a discussion of the physical principles that underlie the fundamental interactions of elementary particle physics and their description by gauge field theories. (shrink)

It is a commonplace in the philosophy of physics that any local physical theory can be represented using arbitrary coordinates, simply by using tensor calculus. On the other hand, the physics literature often claims that spinors \emph{as such} cannot be represented in coordinates in a curved space-time. These commonplaces are inconsistent. What general covariance means for theories with fermions, such as electrons, is thus unclear. In fact both commonplaces are wrong. Though it is not widely known, Ogievetsky and Polubarinov constructed (...) spinors in coordinates in 1965, enhancing the unity of physics and helping to spawn particle physicists' concept of nonlinear group representations. Roughly and locally, these spinors resemble the orthonormal basis or "tetrad" formalism in the symmetric gauge, but they are conceptually self-sufficient and more economical. The typical tetrad formalism is de-Ockhamized, with six extra field components and six compensating gauge symmetries to cancel them out. The Ogievetsky-Polubarinov formalism, by contrast, is Ockhamized, with most of the fluff removed. As developed nonperturbatively by Bilyalov, it admits any coordinates at a point, but "time" must be listed first. Here "time" is defined in terms of an eigenvalue problem involving the metric components and the matrix $diag$, the product of which must have no negative eigenvalues in order to yield a real symmetric square root that is a function of the metric. Thus even formal general covariance requires reconsideration; the atlas of admissible coordinate charts should be sensitive to the types and \emph{values} of the fields involved. Apart from coordinate order and the usual spinorial two-valuedness, Ogievetsky-Polubarinov spinors form, with the metric, a nonlinear geometric object, for which important results on Lie and covariant differentiation are recalled. Such spinors avoid a spurious absolute object in the Anderson-Friedman analysis of substantive general covariance. They also permit the gauge-invariant localization of the infinite-component gravitational energy in General Relativity. Density-weighted spinors exploit the conformal invariance of the massless Dirac equation to show that the volume element is absent. Thus instead of an arbitrary nonsingular matrix with 16 components, 6 of which are gauged away by a new local $O$ gauge group and one of which is irrelevant due to conformal covariance, one can, and presumably should, use density-weighted Ogievetsky-Polubarinov spinors coupled to the 9-component symmetric square root of the part of the metric that fixes null cones. Thus $\frac{7}{16}$ of the orthonormal basis is eliminated as surplus structure. Greater unity between spinors and tensors and the like is achieved, such as regarding conservation laws. Regarding the conventionality of simultaneity, an unusually wide range of $\epsilon$ values is admissible, but some extreme values are inadmissible. Standard simultaneity uniquely makes the spinor transformation law linear and independent of the metric, because transformations among the standard Cartesian coordinate systems fall within the conformal group, for which the spinor transformation law is linear. The surprising mildness of the restrictions on coordinate order as applied to the Schwarzschild solution is exhibited. (shrink)

Barbour, Hawking, Misner and others have argued that time cannot play an essential role in the formulation of a quantum theory of cosmology. Here we present three challenges to their arguments, taken from works and remarks by Kauffman, Markopoulou and Newman. These can be seen to be based on two principles: that every observable in a theory of cosmology should be measurable by some observer inside the universe, and all mathematical constructions necessary to the formulation of the theory should be (...) realizable in a finite time by a computer that fits inside the universe. We also briefly discuss how a cosmological theory could be formulated so it is in agreement with these principles. (shrink)