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Spacetime Algebra
In mathematical physics, spacetime algebra (STA) is a name for the Clifford algebra Cl1,3(R), or equivalently the geometric algebra . According to David Hestenes, spacetime algebra can be particularly closely associated with the geometry of special relativity and relativistic spacetime. It is a vector space that allows not only vectors, but also bivectors (directed quantities associated with particular planes, such as areas, or rotations) or blades (quantities associated with particular hyper-volumes) to be combined, as well as rotated, reflected, or Lorentz boosted. It is also the natural parent algebra of spinors in special relativity. These properties allow many of the most important equations in physics to be expressed in particularly simple forms, and can be very helpful towards a more geometric understanding of their meanings. Structure The spacetime algebra may be built up from an orthogonal basis of one time-like vector \gamma_0 and three space-like vectors, \, with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Physics
Mathematical physics refers to the development of 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 and conserved quantities during the dynamical evoluti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scalar (mathematics)
A scalar is an element of a field which is used to define a ''vector space''. In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through the operation of scalar multiplication (defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector. Generally speaking, a vector space may be defined by using any field instead of real numbers (such as complex numbers). Then scalars of that vector space will be elements of the associated field (such as complex numbers). A scalar product operation – not to be confused with scalar multiplication – may be defined on a vector space, allowing two vectors to be multiplied in the defined way to produce a scalar. A vector space equipped with a scalar product is called an inner product space. A quantity described by multiple scalars, such as having both direction and magnitude, is called a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pauli Equation
In quantum mechanics, the Pauli equation or Schrödinger–Pauli equation is the formulation of the Schrödinger equation for spin-½ particles, which takes into account the interaction of the particle's spin with an external electromagnetic field. It is the non- relativistic limit of the Dirac equation and can be used where particles are moving at speeds much less than the speed of light, so that relativistic effects can be neglected. It was formulated by Wolfgang Pauli in 1927. Equation For a particle of mass m and electric charge q, in an electromagnetic field described by the magnetic vector potential \mathbf and the electric scalar potential \phi, the Pauli equation reads: Here \boldsymbol = (\sigma_x, \sigma_y, \sigma_z) are the Pauli operators collected into a vector for convenience, and \mathbf = -i\hbar \nabla is the momentum operator in position representation. The state of the system, , \psi\rangle (written in Dirac notation), can be considered as a tw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Light-cone
In special and general relativity, a light cone (or "null cone") is the path that a flash of light, emanating from a single event (localized to a single point in space and a single moment in time) and traveling in all directions, would take through spacetime. Details If one imagines the light confined to a two-dimensional plane, the light from the flash spreads out in a circle after the event E occurs, and if we graph the growing circle with the vertical axis of the graph representing time, the result is a cone, known as the future light cone. The past light cone behaves like the future light cone in reverse, a circle which contracts in radius at the speed of light until it converges to a point at the exact position and time of the event E. In reality, there are three space dimensions, so the light would actually form an expanding or contracting sphere in three-dimensional (3D) space rather than a circle in 2D, and the light cone would actually be a four-dimensional version of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero Divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero divisor if there exists a nonzero in such that . This is a partial case of divisibility in rings. An element that is a left or a right zero divisor is simply called a zero divisor. An element that is both a left and a right zero divisor is called a two-sided zero divisor (the nonzero such that may be different from the nonzero such that ). If the ring is commutative, then the left and right zero divisors are the same. An element of a ring that is not a left zero divisor is called left regular or left cancellable. Similarly, an element of a ring that is not a right zero divisor is called right regular or right cancellable. An element of a ring that is left and right cancellable, and is hence not a zero divisor, is called re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Idempotent Element
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of places in abstract algebra (in particular, in the theory of projectors and closure operators) and functional programming (in which it is connected to the property of referential transparency). The term was introduced by American mathematician Benjamin Peirce in 1870 in the context of elements of algebras that remain invariant when raised to a positive integer power, and literally means "(the quality of having) the same power", from + '' potence'' (same + power). Definition An element x of a set S equipped with a binary operator \cdot is said to be ''idempotent'' under \cdot if : . The ''binary operation'' \cdot is said to be ''idempotent'' if : . Examples * In the monoid (\mathbb, \times) of the natural numbers with multiplication, on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Division Algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. Definitions Formally, we start with a non-zero algebra ''D'' over a field. We call ''D'' a division algebra if for any element ''a'' in ''D'' and any non-zero element ''b'' in ''D'' there exists precisely one element ''x'' in ''D'' with ''a'' = ''bx'' and precisely one element ''y'' in ''D'' such that . For associative algebras, the definition can be simplified as follows: a non-zero associative algebra over a field is a division algebra if and only if it has a multiplicative identity element 1 and every non-zero element ''a'' has a multiplicative inverse (i.e. an element ''x'' with ). Associative division algebras The best-known examples of associative division algebras are the finite-dimensional real ones (that is, algebras over the field R of real numbers, which are finite- dimensional as a vector spa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pauli Matrix
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in connection with isospin symmetries. \begin \sigma_1 = \sigma_\mathrm &= \begin 0&1\\ 1&0 \end \\ \sigma_2 = \sigma_\mathrm &= \begin 0& -i \\ i&0 \end \\ \sigma_3 = \sigma_\mathrm &= \begin 1&0\\ 0&-1 \end \\ \end These matrices are named after the physicist Wolfgang Pauli. In quantum mechanics, they occur in the Pauli equation which takes into account the interaction of the spin of a particle with an external electromagnetic field. They also represent the interaction states of two polarization filters for horizontal/vertical polarization, 45 degree polarization (right/left), and circular polarization (right/left). Each Pauli matrix is Hermitian, and together with the ident ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lorentz Factor
The Lorentz factor or Lorentz term is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativity, and it arises in derivations of the Lorentz transformations. The name originates from its earlier appearance in Lorentzian electrodynamics – named after the Dutch physicist Hendrik Lorentz. It is generally denoted (the Greek lowercase letter gamma). Sometimes (especially in discussion of superluminal motion) the factor is written as (Greek uppercase-gamma) rather than . Definition The Lorentz factor is defined as :\gamma = \frac = \frac = \frac , where: *''v'' is the relative velocity between inertial reference frames, *''c'' is the ''speed of light in a vacuum'', * is the ratio of ''v'' to ''c'', *''t'' is coordinate time, * is the proper time for an observer (measuring time intervals in the observer's own frame). This is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Directional Derivative
In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. The directional derivative of a scalar function ''f'' with respect to a vector v at a point (e.g., position) x may be denoted by any of the following: \nabla_(\mathbf)=f'_\mathbf(\mathbf)=D_\mathbff(\mathbf)=Df(\mathbf)(\mathbf)=\partial_\mathbff(\mathbf)=\mathbf\cdot=\mathbf\cdot \frac. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinear coordinate curves, all other coordinates being constant. The directional derivative is a special case of the Gateaux derivative. Definition The ''directional derivative'' of a scalar function :f(\mathbf) = f(x_1, x_2, \ldots, x_n) along a vector :\mathbf = (v_1, \ldots, v_n) is the function \nabla_ defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Einstein Notation
In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving brevity. As part of mathematics it is a notational subset of Ricci calculus; however, it is often used in physics applications that do not distinguish between tangent and cotangent spaces. It was introduced to physics by Albert Einstein in 1916. Introduction Statement of convention According to this convention, when an index variable appears twice in a single term and is not otherwise defined (see Free and bound variables), it implies summation of that term over all the values of the index. So where the indices can range over the set , : y = \sum_^3 c_i x^i = c_1 x^1 + c_2 x^2 + c_3 x^3 is simplified by the convention to: : y = c_i x^i The upper indices are not exponents but are indices ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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