analytical mechanics In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by many scientists and mathematicians during the ...

and

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 ...

, minimal coupling refers to a coupling between

fields Fields may refer to: Music *Fields (band), an indie rock band formed in 2006 *Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song by ...

which involves only the

charge Charge or charged may refer to: Arts, entertainment, and media Films * ''Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * ''Charge!!'', an album by The Aqua ...

distribution and not higher

multipole moments A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, \R^3. Similarly t ...

of the charge distribution. This minimal coupling is in contrast to, for example, Pauli coupling, which includes the

magnetic moment In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagnets ...

of an

electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kn ...

directly in the

Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...

Electrodynamics

electrodynamics In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...

, minimal coupling is adequate to account for all electromagnetic interactions. Higher moments of particles are consequences of minimal coupling and non-zero

spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...

Non-relativistic charged particle in an electromagnetic field

Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...

, the

of a non-relativistic classical particle in an electromagnetic field is (in

SI Units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...

): :

\mathcal = \sum_i \tfrac m \dot_i^2 + \sum_i q \dot_i A_i - q \varphi

where is the

electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...

of the particle, is the

electric scalar potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...

, and the are the components of the

magnetic vector potential In classical electromagnetism, magnetic vector potential (often called A) is the vector quantity defined so that its curl is equal to the magnetic field: \nabla \times \mathbf = \mathbf. Together with the electric potential ''φ'', the magnetic v ...

that may all explicitly depend on

x_i

and

t

. This Lagrangian, combined with

Euler–Lagrange equation In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...

, produces the

Lorentz force In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an elect ...

law :

m \ddot = q \mathbf + q \dot \times \mathbf  \, ,

and is called minimal coupling. Note that the values of scalar potential and vector potential would change during a

gauge transformation In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...

, and the Lagrangian itself will pick up extra terms as well, but the extra terms in the Lagrangian add up to a total time derivative of a scalar function, and therefore still produce the same Euler–Lagrange equation. The

canonical momenta In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example,

gauge invariant In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...

, and are not physically measurable. However, the

kinetic momenta :

P_i \equiv m\dot_i = p_i - q A_i

are gauge invariant and physically measurable. The Hamiltonian mechanics">Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...

, as the Legendre transformation of the Lagrangian, is therefore :

\mathcal = \left\ - \mathcal = \sum_i \frac   + q \varphi

This equation is used frequently in quantum mechanics. Under a gauge transformation, :

\mathbf \rightarrow \mathbf+\nabla f \,, \quad \varphi \rightarrow \varphi-\dot f \,,

where ''f''(r,''t'') is any scalar function of space and time, the aforementioned Lagrangian, canonical momenta and Hamiltonian transform like :

L \rightarrow L'= L+q\frac \,, \quad \mathbf \rightarrow \mathbf = \mathbf+q\nabla f \,, \quad H \rightarrow H' = H-q\frac    \,,

which still produces the same Hamilton's equation: :

\begin
\left.\frac\_&=\left.\frac\_(\dot x_ip'_i-L')=-\left.\frac\_ \\
&=-\left.\frac\_-q\left.\frac\_\frac \\
&= -\frac\left(\left.\frac\_+q\left.\frac\_\right)\\
&=-\dot p'_i
\end

In quantum mechanics, the

wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...

will also undergo a

local Local may refer to: Geography and transportation * Local (train), a train serving local traffic demand * Local, Missouri, a community in the United States * Local government, a form of public administration, usually the lowest tier of administrat ...

U(1) In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers. \mathbb T = \. ...

group transformation during the gauge transformation, which implies that all physical results must be invariant under local U(1) transformations.

Relativistic charged particle in an electromagnetic field

The relativistic Lagrangian for a particle (

rest mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, i ...

and

) is given by: :

\mathcal(t) = - m c^2 \sqrt  + q \dot(t) \cdot \mathbf \left(\mathbf(t),t\right) - q \varphi \left(\mathbf(t),t\right)

Thus the particle's canonical momentum is :

\mathbf(t) = \frac = \frac + q \mathbf

that is, the sum of the kinetic momentum and the potential momentum. Solving for the velocity, we get :

\dot(t) = \frac

So the Hamiltonian is :

\mathcal(t) = \dot \cdot \mathbf - \mathcal = c \sqrt  + q \varphi

This results in the force equation (equivalent to the

) :

\dot = - \frac = q \dot\cdot(\boldsymbol \mathbf) - q \boldsymbol \varphi = q \boldsymbol(\dot \cdot\mathbf)  - q \boldsymbol \varphi

from which one can derive :

\begin
\frac\mathrm\left(\frac \right)
&=\frac\mathrm(\mathbf - q \mathbf)=\dot\mathbf-q\frac-q(\dot\mathbf\cdot\nabla)\mathbf
\\
&=q \boldsymbol(\dot \cdot\mathbf) - q \boldsymbol \varphi -q\frac-q(\dot\mathbf\cdot\nabla)\mathbf \\
&= q \mathbf + q \dot \times \mathbf
\end

The above derivation makes use of the

vector calculus identity The following are important identities involving derivatives and integrals in vector calculus. Operator notation Gradient For a function f(x, y, z) in three-dimensional Cartesian coordinate variables, the gradient is the vector field: \o ...

: :

\tfrac \nabla \left( \mathbf \cdot \mathbf \right)
\ =\  \mathbf \cdot \mathbf_\mathbf \ =\  \mathbf \cdot (\nabla \mathbf) \ =\ (\mathbf  \nabla) \mathbf \,+\, \mathbf  (\nabla  \mathbf) 
.

An equivalent expression for the Hamiltonian as function of the relativistic (kinetic) momentum, , is :

\mathcal(t) = \dot(t) \cdot \mathbf(t) +\frac + q \varphi (\mathbf(t),t)=\gamma mc^2+ q \varphi (\mathbf(t),t)=E+V

This has the advantage that kinetic momentum can be measured experimentally whereas canonical momentum cannot. Notice that the Hamiltonian (

total energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat an ...

) can be viewed as the sum of the relativistic energy (kinetic+rest), , plus the

potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...

, .

Inflation

In studies of

cosmological inflation In physical cosmology, cosmic inflation, cosmological inflation, or just inflation, is a theory of exponential expansion of space in the early universe. The inflationary epoch lasted from seconds after the conjectured Big Bang singularit ...

, ''minimal coupling'' of a scalar field usually refers to minimal coupling to gravity. This means that the action for the

inflaton field The inflaton field is a hypothetical scalar field which is conjectured to have driven cosmic inflation in the very early universe. The field, originally postulated by Alan Guth, provides a mechanism by which a period of rapid expansion from 10&mi ...

\varphi

is not coupled to the

scalar curvature In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry ...

. Its only coupling to gravity is the coupling to the

Lorentz invariant In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of v ...

measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Mea ...

\sqrt\, d^4 x

constructed from the

metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathema ...

(in Planck units): :

S  =\int d^4 x \, \sqrt \, \left(-\fracR + \frac\nabla_\mu \varphi \nabla^\mu \varphi - V(\varphi)\right)

where

g := \det g_

, and utilizing the

gauge covariant derivative The gauge covariant derivative is a variation of the covariant derivative used in general relativity, quantum field theory and fluid dynamics. If a theory has gauge transformations, it means that some physical properties of certain equations are p ...

References

{{DEFAULTSORT:Minimal coupling Gauge theories Hamiltonian mechanics Lagrangian mechanics