Charge (physics)

A charge , a common symbol or , in physics, is a quantity that is defined and interpreted differently depending on the theoretical structure of physics . What all definitions have in common is that in the borderline case of a "subordinate" theory they agree with the definition there. ${\ displaystyle Q}$ ${\ displaystyle q}$

The only charge that plays a role in everyday practical life is the electrical charge . If the term cargo is therefore used without further specification, this cargo is usually meant.

Classical physics

The only charge that occurs in classical physics is the electric charge. It describes the strength with which a particle (physics) interacts with the electric field . The electrical charge is classically identical to a coupling constant between force fields and matter. In classical physics, a moving charge is called (electrical) current ; the current determines how strongly a particle interacts with the magnetic field . Conversely, every charge emits an electric field and every current a magnetic field. When describing the equations of motion for the electric and magnetic field, the Maxwell equations , charge density and current density are included as parameters. The continuity equation follows from Maxwell's equations , which states that charge is a conservation quantity. ${\ displaystyle I}$ ${\ displaystyle \ rho = \ partial Q / \ partial V}$ ${\ displaystyle {\ vec {j}} = \ partial I / \ partial {\ vec {A}}}$

Special theory of relativity

At the time when the special theory of relativity was developed in 1905, only the electric charge was known. Since the Maxwell equations are also relativistically valid equations, the continuity equation can be represented in a relativistically covariant formulation, whereby the charge density is understood as the zeroth component of the four-fold current density: ${\ displaystyle j ^ {\ mu} = {\ begin {pmatrix} c \ rho & {\ vec {j}} \ end {pmatrix}}}$

Non-relativistic quantum mechanics

In non-relativistic quantum mechanics , the charge, as it is the property of a particle in classical physics, is the coupling constant with which a wave function couples to the electrical potential and the vector potential . The formalism used for this is called minimal coupling . Since in quantum mechanics the wave function itself is subject to a continuity equation , according to which the absolute probability of encountering a particle is obtained, the size of the charge associated with the particle is also preserved. This equation of continuity cannot be represented in a relativistically covariant manner.

Quantum field theory

The quantum field theory describes the union of quantum mechanics with the special theory of relativity. In it no longer only force fields are viewed as fields, but also matter. In quantum field theory, the term charge is used twice, on the one hand as a charge operator and on the other hand as its eigenvalue . The charge operator is defined using Noether's theorem . The Noether theorem is also valid in classical mechanics and states that every continuous symmetry of a system has a conservation quantity. For fields it yields a relativistic covariant continuity equation with the charge density ${\ displaystyle {\ hat {Q}}}$ ${\ displaystyle q}$

{\ displaystyle {\ hat {\ rho}} ^ {a} = g \ psi _ {i} ^ {\ dagger} T_ {ij} ^ {a} \ psi _ {j} -f ^ {abc} A ^ {\ nu b} F_ {0 \ nu} ^ {c}}

Here designated

${\ displaystyle \ psi}$ the fermionic field operators,
${\ displaystyle A}$ the vector bosonic field operators,
${\ displaystyle F}$ the field strength tensor
${\ displaystyle T}$ the generators of the symmetry group
${\ displaystyle f}$ the structural constants of the symmetry group
${\ displaystyle g}$ a coupling constant

The charge operator exchanges with the Hamilton operator , so a common eigenbase can be chosen so that the observable particles always have a well-defined charge as eigenvalue to the operator. In particular, the definition of the charge operator contains the coupling constant, but charge and coupling constant are different objects.

In unbroken theories, the charge operator annihilates the vacuum; denotes the quantum vacuum, is . This is not the case in theories with spontaneously broken symmetry; it grips the Fabri-Picasso's theorem , according to which the operator norm of the charge operator is infinite: . The definition of a charge as an eigenvalue of such an operator is not possible. Therefore, in the Standard Model, there is both a strong charge, called a color charge , which can be assigned to the, and an electrical charge which results from the refraction of the , but not a “weak charge”. ${\ displaystyle | \ Omega \ rangle}$ ${\ displaystyle {\ hat {Q}} | \ Omega \ rangle = 0}$ ${\ displaystyle \ | {\ hat {Q}} \ | = \ infty}$ ${\ displaystyle SU (3) _ {C}}$ ${\ displaystyle SU (2) _ {L} \ otimes U (1) _ {Y}}$ ${\ displaystyle U (1) _ {Q}}$

In the case of non-Abelian symmetry groups like that , these Noether charges are still retained, but corrections of higher orders to their eigenvalues are no longer gauge-invariant. As in the classic case, the charge is determined from the potential with the aid of the coupling constant for these corrections. The violation of the gauge invariance is a direct consequence of the Weinberg-Witten theorem . ${\ displaystyle SU (3)}$

Mass as charge

The mass is separated from the perspective of classical physics semantically from the concept of charge, but may charge as the gravitational be construed. A quantity that obeys a continuity equation according to Noether's theorem is the energy-momentum tensor , whose component in the classical Limes corresponds to the mass density. ${\ displaystyle T ^ {\ mu \ nu}}$ ${\ displaystyle T ^ {00}}$

literature

Ian JR Aitchison and Anthony JG Hey: Gauge Theories in Particle Physics, Volume 2: Non-Abelian Gauge Theories: QCD and the Electroweak Theory . 3. Edition. Institute of Physics Publishing, Bristol, Philadelphia 2004, ISBN 0-7503-0950-4 (English).
Lewis H. Ryder: Quantum Field Theory . 2nd Edition. Cambridge University Press, Cambridge 1996, ISBN 0-521-47242-3 (English).
Mattew D. Schwartz: Quantum Field Theory and the Standard Model . 1st edition. Cambridge University Press, Cambridge 2014, ISBN 978-1-107-03473-0 (English).