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Dirac Equation In Curved Spacetime
In mathematical physics, the Dirac equation in curved spacetime is a generalization of the Dirac equation from flat spacetime (Minkowski space) to curved spacetime, a general Lorentzian manifold. Mathematical formulation Spacetime In full generality the equation can be defined on M or (M,\mathbf) a pseudo-Riemannian manifold, but for concreteness we restrict to pseudo-Riemannian manifold with signature (- + + +). The metric is referred to as \mathbf, or g_ in abstract index notation. Frame fields We use a set of vierbein or frame fields \ = \, which are a set of vector fields (which are not necessarily defined globally on M). Their defining equation is :g_e_\mu^a e_\nu^b = \eta_. The vierbein defines a local rest frame, allowing the constant Gamma matrices to act at each spacetime point. In differential-geometric language, the vierbein is equivalent to a section of the frame bundle, and so defines a local trivialization of the frame bundle. Spin connection To wr ...
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Mathematical Physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". An alternative definition would also include those mathematics that are inspired by physics (also known as physical mathematics). Scope There are several distinct branches of mathematical physics, and these roughly correspond to particular historical periods. Classical mechanics The rigorous, abstract and advanced reformulation of Newtonian mechanics adopting the Lagrangian mechanics and the Hamiltonian mechanics even in the presence of constraints. Both formulations are embodied in analytical mechanics and lead to understanding the deep interplay of the notions of symmetry (physics), symmetry and conservation law, con ...
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Erwin Schrödinger
Erwin Rudolf Josef Alexander Schrödinger (, ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was a Nobel Prize-winning Austrian physicist with Irish citizenship who developed a number of fundamental results in quantum theory: the Schrödinger equation provides a way to calculate the wave function of a system and how it changes dynamically in time. In addition, he wrote many works on various aspects of physics: statistical mechanics and thermodynamics, physics of dielectrics, colour theory, electrodynamics, general relativity, and cosmology, and he made several attempts to construct a unified field theory. In his book ''What Is Life?'' Schrödinger addressed the problems of genetics, looking at the phenomenon of life from the point of view of physics. He also paid great attention to the philosophical aspects of science, ancient, and oriental philosophical concepts, ethics, and religion. He also wrote on philosophy and theoretical biology. In popular culture, ...
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Partial Differential Equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity, and stability. Among the many open questions are the e ...
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Spinors
In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight (infinitesimal) rotation. Unlike vectors and tensors, a spinor transforms to its negative when the space is continuously rotated through a complete turn from 0° to 360° (see picture). This property characterizes spinors: spinors can be viewed as the "square roots" of vectors (although this is inaccurate and may be misleading; they are better viewed as "square roots" of sections of vector bundles – in the case of the exterior algebra bundle of the cotangent bundle, they thus become "square roots" of differential forms). It is also possible to associate a substantially similar notion of spinor to Minkowski space, in which case the Lorentz transformations of special relativity play the role of rotations. Spinors were introduced in geome ...
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Quantum Field Theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles. QFT treats particles as excited states (also called Quantum, quanta) of their underlying quantum field (physics), fields, which are more fundamental than the particles. The equation of motion of the particle is determined by minimization of the Lagrangian, a functional of fields associated with the particle. Interactions between particles are described by interaction terms in the Lagrangian (field theory), Lagrangian involving their corresponding quantum fields. Each interaction can be visually represented by Feynman diagrams according to perturbation theory (quantum mechanics), perturbation theory in quantum mechanics. History Quantum field theory emerged from the wo ...
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Two-body Dirac Equations
In quantum field theory, and in the significant subfields of quantum electrodynamics (QED) and quantum chromodynamics (QCD), the two-body Dirac equations (TBDE) of constraint dynamics provide a three-dimensional yet manifestly covariant reformulation of the Bethe–Salpeter equation for two spin-1/2 particles. Such a reformulation is necessary since without it, as shown by Nakanishi, the Bethe–Salpeter equation possesses negative-norm solutions arising from the presence of an essentially relativistic degree of freedom, the relative time. These "ghost" states have spoiled the naive interpretation of the Bethe–Salpeter equation as a quantum mechanical wave equation. The two-body Dirac equations of constraint dynamics rectify this flaw. The forms of these equations can not only be derived from quantum field theory they can also be derived purely in the context of Dirac's constraint dynamics P.A.M. Dirac, Lectures on Quantum Mechanics (Yeshiva University, New York, 1964) and relat ...
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Maxwell's Equations In Curved Spacetime
In physics, Maxwell's equations in curved spacetime govern the dynamics of the electromagnetic field in curved spacetime (where the metric may not be the Minkowski metric) or where one uses an arbitrary (not necessarily Cartesian) coordinate system. These equations can be viewed as a generalization of the vacuum Maxwell's equations which are normally formulated in the local coordinates of flat spacetime. But because general relativity dictates that the presence of electromagnetic fields (or energy/matter in general) induce curvature in spacetime, Maxwell's equations in flat spacetime should be viewed as a convenient approximation. When working in the presence of bulk matter, distinguishing between free and bound electric charges may facilitate analysis. When the distinction is made, they are called the macroscopic Maxwell's equations. Without this distinction, they are sometimes called the "microscopic" Maxwell's equations for contrast. The electromagnetic field admits a coor ...
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Dirac Spinor
In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos. It appears in the plane-wave solution to the Dirac equation, and is a certain combination of two Weyl spinors, specifically, a bispinor that transforms "spinorially" under the action of the Lorentz group. Dirac spinors are important and interesting in numerous ways. Foremost, they are important as they do describe all of the known fundamental particle fermions in nature; this includes the electron and the quarks. Algebraically they behave, in a certain sense, as the "square root" of a vector. This is not readily apparent from direct examination, but it has slowly become clear over the last 60 years that spinorial representations are fundamental to geometry. For example, effectively all Riemannian manifolds can have spinors and spin connections built upon them, via the Clifford algebra. The Dirac spinor is specific to that ...
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Dirac Equation In The Algebra Of Physical Space
In physics, the algebra of physical space (APS) is the use of the Clifford algebra, Clifford or geometric algebra Cl3,0(R) of the three-dimensional Euclidean space as a model for (3+1)-dimensional spacetime, representing a point in spacetime via a paravector (3-dimensional vector plus a 1-dimensional scalar). The Clifford algebra Cl3,0(R) has a faithful representation, generated by Pauli matrices, on the spin representation C2; further, Cl3,0(R) is isomorphic to the even subalgebra Cl(R) of the Clifford algebra Cl3,1(R). APS can be used to construct a compact, unified and geometrical formalism for both classical and quantum mechanics. APS should not be confused with spacetime algebra (STA), which concerns the Clifford algebra Cl1,3(R) of the four-dimensional Minkowski spacetime. Special relativity Spacetime position paravector In APS, the spacetime position is represented as the paravector x = x^0 + x^1 \mathbf_1 + x^2 \mathbf_2 + x^3 \mathbf_3, where the time is given by the ...
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Volume Form
In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of the line bundle \textstyle^n(T^*M), denoted as \Omega^n(M). A manifold admits a nowhere-vanishing volume form if and only if it is orientable. An orientable manifold has infinitely many volume forms, since multiplying a volume form by a function yields another volume form. On non-orientable manifolds, one may instead define the weaker notion of a density. A volume form provides a means to define the integral of a function on a differentiable manifold. In other words, a volume form gives rise to a measure with respect to which functions can be integrated by the appropriate Lebesgue integral. The absolute value of a volume form is a volume element, which is also known variously as a ''twisted volume form'' or ''pseudo-volume form''. It als ...
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Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to associate pictures with coordinates (e.g ...
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Lorentz Covariance
In relativistic physics, Lorentz symmetry or Lorentz invariance, named after the Dutch physicist Hendrik Lorentz, is an equivalence of observation or observational symmetry due to special relativity implying that the laws of physics stay the same for all observers that are moving with respect to one another within an inertial frame. It has also been described as "the feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through space". Lorentz covariance, a related concept, is a property of the underlying spacetime manifold. Lorentz covariance has two distinct, but closely related meanings: # A physical quantity is said to be Lorentz covariant if it transforms under a given representation of the Lorentz group. According to the representation theory of the Lorentz group, these quantities are built out of scalars, four-vectors, four-tensors, and spinors. In particular, a Lorentz covariant scalar (e.g., the spa ...
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