Riemann–Silberstein Vector
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In
mathematical physics Mathematical physics is 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 de ...
, in particular
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
, the Riemann–Silberstein vector or Weber vector named after
Bernhard Riemann Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the f ...
,
Heinrich Martin Weber Heinrich Martin Weber (5 March 1842, Heidelberg, German Confederation, Germany – 17 May 1913, Straßburg, Alsace-Lorraine, German Empire, now Strasbourg, France) was a German mathematician. Weber's main work was in algebra, number theory, ...
and
Ludwik Silberstein Ludwik Silberstein (May 17, 1872 – January 17, 1948) was a Polish-American physicist who helped make special relativity and general relativity staples of university coursework. His textbook '' The Theory of Relativity'' was published by Macmill ...
, (or sometimes ambiguously called the "electromagnetic field") is a
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
vector Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...
that combines the
electric field An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...
E and the
magnetic field A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular ...
B.


History

Heinrich Martin Weber Heinrich Martin Weber (5 March 1842, Heidelberg, German Confederation, Germany – 17 May 1913, Straßburg, Alsace-Lorraine, German Empire, now Strasbourg, France) was a German mathematician. Weber's main work was in algebra, number theory, ...
published the fourth edition of "The
partial differential equations In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how ...
of mathematical physics according to Riemann's lectures" in two volumes (1900 and 1901). However, Weber pointed out in the preface of the first volume (1900) that this fourth edition was completely rewritten based on his own lectures, not Riemann's, and that the reference to "Riemann's lectures" only remained in the title because the overall concept remained the same and that he continued the work in Riemann's spirit. In the second volume (1901, §138, p. 348), Weber demonstrated how to consolidate Maxwell's equations using \mathfrak + i\ \mathfrak. The real and imaginary components of the equation :\operatorname(\mathfrak + i\ \mathfrak) = \frac\ \frac are an interpretation of Maxwell's equations without charges or currents. It was independently rediscovered and further developed by
Ludwik Silberstein Ludwik Silberstein (May 17, 1872 – January 17, 1948) was a Polish-American physicist who helped make special relativity and general relativity staples of university coursework. His textbook '' The Theory of Relativity'' was published by Macmill ...
in 1907.


Definition

Given an electric field E and a magnetic field B defined on a common
region In geography, regions, otherwise referred to as areas, zones, lands or territories, are portions of the Earth's surface that are broadly divided by physical characteristics (physical geography), human impact characteristics (human geography), and ...
of
spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
, the Riemann–Silberstein vector is \mathbf = \mathbf + ic \mathbf , where is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
, with some authors preferring to multiply the right hand side by an overall constant \sqrt, where is the
permittivity of free space Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...
. It is analogous to the
electromagnetic tensor In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. Th ...
''F'', a 2-vector used in the
covariant formulation of classical electromagnetism The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations and the Lorentz force) in a form that is manifestly invariant under Lorentz transform ...
. In Silberstein's formulation, ''i'' was defined as the
imaginary unit The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...
, and F was defined as a complexified 3-dimensional
vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
, called a ''
bivector In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. Considering a scalar as a degree-zero quantity and a vector as a degree-one quantity, a bivector is of ...
field''.


Application

The Riemann–Silberstein vector is used as a point of reference in the geometric algebra formulation of electromagnetism. Maxwell's ''four'' equations in
vector calculus Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, \mathbb^3. The term ''vector calculus'' is sometimes used as a ...
reduce to ''one'' equation in the
algebra of physical space Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
: : \left(\frac\dfrac + \boldsymbol\right)\mathbf = \frac\left( \rho - \frac\mathbf \right). Expressions for the fundamental invariants and the
energy density In physics, energy density is the quotient between the amount of energy stored in a given system or contained in a given region of space and the volume of the system or region considered. Often only the ''useful'' or extractable energy is measure ...
and
momentum In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. ...
density also take on simple forms: : \mathbf^2 = \mathbf^2 - c^2\mathbf^2 + 2 i c\mathbf \cdot \mathbf : \frac\mathbf^ \mathbf = \frac\left( \mathbf^2 + c^2\mathbf^2 \right) + \frac \mathbf, where S is the
Poynting vector In physics, the Poynting vector (or Umov–Poynting vector) represents the directional energy flux (the energy transfer per unit area, per unit time) or '' power flow'' of an electromagnetic field. The SI unit of the Poynting vector is the wat ...
. The Riemann–Silberstein vector is used for an exact matrix representations of Maxwell's equations in an inhomogeneous medium with sources.


Photon wave function

In 1996 contribution to
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the Theory of relativity, relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quant ...
, Iwo Bialynicki-Birula used the Riemann–Silberstein vector as the basis for an approach to the
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...
, noting that it is a "complex vector-function of space coordinates r and time ''t'' that adequately describes the
quantum state In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system ...
of a single photon". To put the Riemann–Silberstein vector in contemporary parlance, a transition is made: :With the advent of
spinor In geometry and physics, spinors (pronounced "spinner" IPA ) are elements of a complex numbers, complex vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infi ...
calculus that superseded the quaternionic calculus, the transformation properties of the Riemann-Silberstein vector have become even more transparent ... a symmetric second-rank spinor. Bialynicki-Birula acknowledges that the photon wave function is a controversial concept and that it cannot have all the properties of Schrödinger
wave function In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
s of non-relativistic wave mechanics. Yet defense is mounted on the basis of practicality: it is useful for describing quantum states of excitation of a free field, electromagnetic fields acting on a medium, vacuum excitation of virtual positron-electron pairs, and presenting the photon among quantum particles that do have wave functions.


Schrödinger equation for the photon and the Heisenberg uncertainty relations

Multiplying the two time dependent Maxwell equations by \hbar the Schrödinger equation for photon in the vacuum is given by :i \hbar \partial_ =c (\mathbf S \cdot \nabla) \mathbf F = c (\mathbf S \cdot \mathbf p ) \mathbf F where is the vector built from the
spin Spin or spinning most often refers to: * Spin (physics) or particle spin, a fundamental property of elementary particles * Spin quantum number, a number which defines the value of a particle's spin * Spinning (textiles), the creation of yarn or thr ...
of the length 1
matrices Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the ...
generating full infinitesimal rotations of 3-spinor particle. One may therefore notice that the Hamiltonian in the Schrödinger equation of the photon is the projection of its spin 1 onto its momentum since the normal momentum operator appears there from combining parts of rotations. In contrast to the electron wave function the modulus square of the wave function of the photon (Riemann-Silbertein vector) is not dimensionless and must be multiplied by the "local photon wavelength" with the proper power to give dimensionless expression to normalize i.e. it is normalized in the exotic way with the integral kernel :\, \mathbf F\, =\int dx^3 dx'^3=1 The two residual Maxwell equations are only constraints i.e. :\nabla \cdot \mathbf F=0 and they are automatically fulfilled all time if only fulfilled at the initial time t=0, i.e. :\mathbf F(0)= \nabla \times \mathbf G where \mathbf G is any complex
vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
with the non-vanishing
rotation Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...
, or it is a vector potential for the Riemann–Silberstein vector. While having the wave function of the photon one can estimate the uncertainty relations for the photon.- This publication is using slightly different definitions of position and momentum uncertainties resigning from the position operator and normalizing uncertainty of r^2 to uncertainty of r It shows up that photons are "more quantum" than the electron while their uncertainties of position and the momentum are higher. The natural candidates to estimate the uncertainty are the natural momentum like simply the projection E/c or H/c from Einstein formula for the photoelectric effect and the simplest theory of quanta and the r, the uncertainty of the position length vector. We will use the general relation for the uncertainty for the operators A, B : \sigma_\sigma_ \geq \frac\left, \langle hat,\hatrangle \. We want the uncertainty relation for \sigma_\sigma_ i.e. for the operators : r^2=x^2+y^2+z^2 : p^2=(\mathbf S \cdot \mathbf p )^2 The first step is to find the auxiliary operator \tilde r such that this relation can be used directly. First we make the same trick for r that Dirac made to calculate the square root of the Klein-Gordon operator to get the
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles, called "Dirac ...
: : \tilde r = \alpha_1 x + \alpha_2 y + \alpha_3 z where \alpha_i are matrices from the Dirac equation: : \alpha_i^2=1 : \alpha_i \alpha_k + \alpha_k \alpha_i= 2 \delta_ Therefore, we have : \tilde r^2 = r^2 Because the spin matrices 1 are only 3 \times 3 to calculate the commutator in the same space we approximate the spin matrices by
angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
matrices of the particle with the length 3/2 \approx 1 while dropping the multiplying 1/2 since the resulting Maxwell equations in 4 dimensions would look too artificial to the original (alternatively we can keep the original 1/2 factors but normalize the new 4-spinor to 2 as 4 scalar particles normalized to 1/2): : \tilde p^2 = (\mathbf \tilde L \cdot \mathbf p)^2 We can now readily calculate the commutator while calculating commutators of \alpha_i matrixes and scaled \tilde L_i and noticing that the symmetric Gaussian state e^ is annihilating in average the terms containing mixed variable like x p_y. Calculating 9 commutators (mixed may be zero by Gaussian example and the L_z \alpha_z=\alpha_z L_z=0 since those matrices are counter-diagonal) and estimating terms from the norm of the resulting 4 \times 4 matrix containing four 2\sqrt 3 factors giving square of the most natural L2, 1 norm of this matrix as 48 \approx 49=7^2 \approx 8^2 and using the norm inequality for the estimate :\lVert\mathbf A \mathbf x\rVert \leq \lVert\mathbf A\rVert \lVert\mathbf x\rVert \approx \lVert\mathbf A\rVert \lVert\mathbf x\rVert we obtain :\left, \langle tilde r,\tilde prangle \\geq 8 \hbar . or :\sigma_\sigma_\geq 4 \hbar which is much more than for the mass particle in 3 dimensions that is :\sigma_\sigma_\geq \frac\hbar and therefore photons turn out to be particles 8/3 times or almost 3 times "more quantum" than particles with the mass like electrons.


See also

* Matrix representation of Maxwell's equations


References

{{DEFAULTSORT:Riemann-Silberstein vector Electromagnetism Geometric algebra Bernhard Riemann