Conservation law

From Wikipedia, the free encyclopedia

In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, conservation of angular momentum, and conservation of electric charge. There are also many approximate conservation laws, which apply to such quantities as mass, parity, lepton number, baryon number, strangeness, hypercharge, etc. These quantities are conserved in certain classes of physics processes, but not in all.

A local conservation law is usually expressed mathematically as a continuity equation, a partial differential equation which gives a relation between the amount of the quantity and the "transport" of that quantity. It states that the amount of the conserved quantity at a point or within a volume can only change by the amount of the quantity which flows in or out of the volume.

From Noether's theorem, each conservation law is associated with a symmetry in the underlying physics.

Conservation laws as fundamental laws of nature

Conservation laws are fundamental to our understanding of the physical world, in that they describe which processes can or cannot occur in nature. For example, the conservation law of energy states that the total quantity of energy in an isolated system does not change, though it may change form. In general, the total quantity of the property governed by that law remains unchanged during physical processes. With respect to classical physics, conservation laws include conservation of energy, mass (or matter), linear momentum, angular momentum, and electric charge. With respect to particle physics, particles cannot be created or destroyed except in pairs, where one is ordinary and the other is an antiparticle. With respect to symmetries and invariance principles, three special conservation laws have been described, associated with inversion or reversal of space, time, and charge.

Conservation laws are considered to be fundamental laws of nature, with broad application in physics, as well as in other fields such as chemistry, biology, geology, and engineering.

Most conservation laws are exact, or absolute, in the sense that they apply to all possible processes. Some conservation laws are partial, in that they hold for some processes but not for others.

One particularly important result concerning conservation laws is Noether's theorem, which states that there is a one-to-one correspondence between each one of them and a differentiable symmetry of nature. For example, the conservation of energy follows from the time-invariance of physical systems, and the conservation of angular momentum arises from the fact that physical systems behave the same regardless of how they are oriented in space.

Exact laws

A partial listing of physical conservation equations due to symmetry that are said to be exact laws, or more precisely have never been proven to be violated:

Conservation Law	Respective Noether symmetry invariance		Number of dimensions
Conservation of mass-energy	Time invariance	Lorentz invariance symmetry	1	translation about time axis
Conservation of linear momentum	Translation symmetry		3	translation about x,y,z position
Conservation of angular momentum	Rotation invariance		3	rotation about x,y,z axes
CPT symmetry (combining charge, parity and time conjugation)	Lorentz invariance		1 + 1 + 1	(charge inversion q → −q) + (position inversion r → −r) + (time inversion t → −t)
Conservation of electric charge	Gauge invariance		1⊗4	scalar field (1D) in 4D spacetime (x,y,z + time evolution)
Conservation of color charge	SU(3) Gauge invariance		3	r,g,b
Conservation of weak isospin	SU(2)L Gauge invariance		1	weak charge
Conservation of probability	Probability invariance[1]		1 ⊗ 4	total probability always = 1 in whole x,y,z space, during time evolution

Approximate laws

There are also approximate conservation laws. These are approximately true in particular situations, such as low speeds, short time scales, or certain interactions.

• Conservation of rest mass
• Conservation of baryon number
• Conservation of lepton number (In the Standard Model)
• Conservation of flavor (violated by the weak interaction)
• Conservation of parity
• Invariance under charge conjugation
• Invariance under time reversal
• CP symmetry, the combination of charge and parity conjugation (equivalent to time reversal if CPT holds)

Global and local conservation laws

The total amount of some conserved quantity in the universe could remain unchanged if an equal amount were to appear at one point A and simultaneously disappear from another separate point B. For example, an amount of energy could appear on Earth without changing the total amount in the Universe if the same amount of energy were to disappear from a remote region of the Universe. This weak form of "global" conservation is really not a conservation law because it is not Lorentz invariant, so phenomena like the above do not occur in nature.^[2][3] Due to Special Relativity, if the appearance of the energy at A and disappearance of the energy at B are simultaneous in one inertial reference frame, they will not be simultaneous in other inertial reference frames moving with respect to the first. In a moving frame one will occur before the other; either the energy at A will appear before or after the energy at B disappears. In both cases, during the interval energy will not be conserved.

A stronger form of conservation law requires that, for the amount of a conserved quantity at a point to change, there must be a flow, or flux of the quantity into or out of the point. For example, the amount of electric charge in a volume is never found to change without an electric current into or out of the volume that carries the difference in charge. Since it only involves continuous local changes, this stronger type of conservation law is Lorentz invariant; a quantity conserved in one reference frame is conserved in all moving reference frames.^[2][3] This is called a local conservation law.^[2][3] Local conservation also implies global conservation; that the total amount of the conserved quantity in the Universe remains constant. All of the conservation laws listed above are local conservation laws. A local conservation law is expressed mathematically by a continuity equation, which states that the change in the quantity in a volume is equal to the total net "flux" of the quantity through the surface of the volume. The following sections discuss continuity equations in general.

Differential forms

In continuum mechanics, the most general form of an exact conservation law is given by a continuity equation. For example, conservation of electric charge q is

${\displaystyle {\frac {\partial \rho }{\partial t}}=-\nabla \cdot \mathbf {j} \,}$

where ∇⋅ is the divergence operator, ρ is the density of q (amount per unit volume), j is the flux of q (amount crossing a unit area in unit time), and t is time.

If we assume that the motion u of the charge is a continuous function of position and time, then

${\displaystyle \mathbf {j} =\rho \mathbf {u} }$

${\displaystyle {\frac {\partial \rho }{\partial t}}=-\nabla \cdot (\rho \mathbf {u} )\,.}$

In one space dimension this can be put into the form of a homogeneous first-order quasilinear hyperbolic equation:^[4]

${\displaystyle y_{t}+A(y)y_{x}=0}$

where the dependent variable y is called the density of a conserved quantity, and A(y) is called the current jacobian, and the subscript notation for partial derivatives has been employed. The more general inhomogeneous case:

${\displaystyle y_{t}+A(y)y_{x}=s}$

is not a conservation equation but the general kind of balance equation describing a dissipative system. The dependent variable y is called a nonconserved quantity, and the inhomogeneous term s(y,x,t) is the-source, or dissipation. For example, balance equations of this kind are the momentum and energy Navier-Stokes equations, or the entropy balance for a general isolated system.

In the one-dimensional space a conservation equation is a first-order quasilinear hyperbolic equation that can be put into the advection form:

${\displaystyle y_{t}+a(y)y_{x}=0}$

where the dependent variable y(x,t) is called the density of the conserved (scalar) quantity (c.q.(d.) = conserved quantity (density)), and a(y) is called the current coefficient, usually corresponding to the partial derivative in the conserved quantity of a current density (c.d.) of the conserved quantity j(y):^[4]

${\displaystyle a(y)=j_{y}(y)}$

In this case since the chain rule applies:

${\displaystyle j_{x}=j_{y}(y)y_{x}=a(y)y_{x}}$

the conservation equation can be put into the current density form:

${\displaystyle y_{t}+j_{x}(y)=0}$

In a space with more than one dimension the former definition can be extended to an equation that can be put into the form:

${\displaystyle y_{t}+\mathbf {a} (y)\cdot \nabla y=0}$

where the conserved quantity is y(r,t), ${\displaystyle \cdot }$ denotes the scalar product, ∇ is the nabla operator, here indicating a gradient, and a(y) is a vector of current coefficients, analogously corresponding to the divergence of a vector c.d. associated to the c.q. j(y):

${\displaystyle y_{t}+\nabla \cdot \mathbf {j} (y)=0}$

This is the case for the continuity equation:

${\displaystyle \rho _{t}+\nabla \cdot (\rho \mathbf {u} )=0}$

Here the conserved quantity is the mass, with density ρ(r,t) and current density ρu, identical to the momentum density, while u(r,t) is the flow velocity.

In the general case a conservation equation can be also a system of this kind of equations (a vector equation) in the form:^[4]

${\displaystyle \mathbf {y} _{t}+\mathbf {A} (\mathbf {y} )\cdot \nabla \mathbf {y} =\mathbf {0} }$

where y is called the conserved (vector) quantity, ∇ y is its gradient, 0 is the zero vector, and A(y) is called the Jacobian of the current density. In fact as in the former scalar case, also in the vector case A(y) usually corresponding to the Jacobian of a current density matrix J(y):

${\displaystyle \mathbf {A} (\mathbf {y} )=\mathbf {J} _{\mathbf {y} }(\mathbf {y} )}$

and the conservation equation can be put into the form:

${\displaystyle \mathbf {y} _{t}+\nabla \cdot \mathbf {J} (\mathbf {y} )=\mathbf {0} }$

For example, this the case for Euler equations (fluid dynamics). In the simple incompressible case they are:

${\displaystyle {\begin{aligned}\nabla \cdot {\mathbf {u} }=0\\[1.2ex]{\partial {\mathbf {u} } \over \partial t}+{\mathbf {u} }\cdot \nabla {\mathbf {u} }+\nabla s={\mathbf {0} },\end{aligned}}}$

where:

• u is the flow velocity vector, with components in a N-dimensional space u₁, u₂ ... u_N,
• s is the specific pressure (pressure per unit density) giving the source term,

It can be shown that the conserved (vector) quantity and the c.d. matrix for these equations are respectively:

${\displaystyle {\mathbf {y} }={\begin{pmatrix}1\\{\mathbf {u} }\end{pmatrix}};\qquad {\mathbf {J} }={\begin{pmatrix}{\mathbf {u} }\\{\mathbf {u} }\otimes {\mathbf {u} }+s{\mathbf {I} }\end{pmatrix}};\qquad }$

where ${\displaystyle \otimes }$ denotes the outer product.

Integral and weak forms

Conservation equations can be also expressed in integral form: the advantage of the latter is substantially that it requires less smoothness of the solution, which paves the way to weak form, extending the class of admissible solutions to include discontinuous solutions.^[5] By integrating in any space-time domain the current density form in 1-D space:

${\displaystyle y_{t}+j_{x}(y)=0}$

and by using Green's theorem, the integral form is:

${\displaystyle \int _{-\infty }^{\infty }y\,dx+\int _{0}^{\infty }j(y)\,dt=0}$

In a similar fashion, for the scalar multidimensional space, the integral form is:

${\displaystyle \oint [y\,d^{N}r+j(y)\,dt]=0}$

where the line integration is performed along the boundary of the domain, in an anticlock-wise manner.^[5]

Moreover, by defining a test function φ(r,t) continuously differentiable both in time and space with compact support, the weak form can be obtained pivoting on the initial condition. In 1-D space it is:

${\displaystyle \int _{0}^{\infty }\int _{-\infty }^{\infty }\phi _{t}y+\phi _{x}j(y)\,dx\,dt=-\int _{-\infty }^{\infty }\phi (x,0)y(x,0)\,dx}$

Note that in the weak form all the partial derivatives of the density and current density have been passed on to the test function, which with the former hypothesis is sufficiently smooth to admit these derivatives.^[5]

Symmetry (physics)

From Wikipedia, the free encyclopedia

 

First Brillouin zone of FCC lattice showing symmetry labels

In physics, a symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation.

A family of particular transformations may be continuous (such as rotation of a circle) or discrete (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups (see Symmetry group).

These two concepts, Lie and finite groups, are the foundation for the fundamental theories of modern physics. Symmetries are frequently amenable to mathematical formulations such as group representations and can, in addition, be exploited to simplify many problems.

Arguably the most important example of a symmetry in physics is that the speed of light has the same value in all frames of reference, which is known in mathematical terms as Poincaré group, the symmetry group of special relativity. Another important example is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations, which is an important idea in general relativity.

Symmetry as a kind of invariance

Invariance is specified mathematically by transformations that leave some property (e.g. quantity) unchanged. This idea can apply to basic real-world observations. For example, temperature may be homogeneous throughout a room. Since the temperature does not depend on the position of an observer within the room, we say that the temperature is invariant under a shift in an observer's position within the room.

Similarly, a uniform sphere rotated about its center will appear exactly as it did before the rotation. The sphere is said to exhibit spherical symmetry. A rotation about any axis of the sphere will preserve how the sphere "looks".

Invariance in force

The above ideas lead to the useful idea of invariance when discussing observed physical symmetry; this can be applied to symmetries in forces as well.

For example, an electric field due to an electrically charged wire of infinite length is said to exhibit cylindrical symmetry, because the electric field strength at a given distance r from the wire will have the same magnitude at each point on the surface of a cylinder (whose axis is the wire) with radius r. Rotating the wire about its own axis does not change its position or charge density, hence it will preserve the field. The field strength at a rotated position is the same. This is not true in general for an arbitrary system of charges.

In Newton's theory of mechanics, given two bodies, each with mass m, starting at the origin and moving along the x-axis in opposite directions, one with speed v₁ and the other with speed v₂ the total kinetic energy of the system (as calculated from an observer at the origin) is ​¹⁄₂m(v₁² + v₂²) and remains the same if the velocities are interchanged. The total kinetic energy is preserved under a reflection in the y-axis.

The last example above illustrates another way of expressing symmetries, namely through the equations that describe some aspect of the physical system. The above example shows that the total kinetic energy will be the same if v₁ and v₂ are interchanged.

Local and global symmetries

Symmetries may be broadly classified as global or local. A global symmetry is one that holds at all points of spacetime, whereas a local symmetry is one that has a different symmetry transformation at different points of spacetime; specifically a local symmetry transformation is parameterised by the spacetime co-ordinates. Local symmetries play an important role in physics as they form the basis for gauge theories.

Continuous symmetries

The two examples of rotational symmetry described above - spherical and cylindrical - are each instances of continuous symmetry. These are characterised by invariance following a continuous change in the geometry of the system. For example, the wire may be rotated through any angle about its axis and the field strength will be the same on a given cylinder. Mathematically, continuous symmetries are described by continuous or smooth functions. An important subclass of continuous symmetries in physics are spacetime symmetries.

Spacetime symmetries

Continuous spacetime symmetries are symmetries involving transformations of space and time. These may be further classified as spatial symmetries, involving only the spatial geometry associated with a physical system; temporal symmetries, involving only changes in time; or spatio-temporal symmetries, involving changes in both space and time.

• Time translation: A physical system may have the same features over a certain interval of time ${\displaystyle \delta t}$; this is expressed mathematically as invariance under the transformation ${\displaystyle t\,\rightarrow t+a}$ for any real numbers t and a in the interval. For example, in classical mechanics, a particle solely acted upon by gravity will have gravitational potential energy ${\displaystyle \,mgh}$ when suspended from a height ${\displaystyle h}$ above the Earth's surface. Assuming no change in the height of the particle, this will be the total gravitational potential energy of the particle at all times. In other words, by considering the state of the particle at some time (in seconds) ${\displaystyle t_{0}}$ and also at ${\displaystyle t_{0}+3}$, say, the particle's total gravitational potential energy will be preserved.
• Spatial translation: These spatial symmetries are represented by transformations of the form ${\displaystyle {\vec {r}}\,\rightarrow {\vec {r}}+{\vec {a}}}$ and describe those situations where a property of the system does not change with a continuous change in location. For example, the temperature in a room may be independent of where the thermometer is located in the room.
• Spatial rotation: These spatial symmetries are classified as proper rotations and improper rotations. The former are just the 'ordinary' rotations; mathematically, they are represented by square matrices with unit determinant. The latter are represented by square matrices with determinant −1 and consist of a proper rotation combined with a spatial reflection (inversion). For example, a sphere has proper rotational symmetry. Other types of spatial rotations are described in the article Rotation symmetry.
• Poincaré transformations: These are spatio-temporal symmetries which preserve distances in Minkowski spacetime, i.e. they are isometries of Minkowski space. They are studied primarily in special relativity. Those isometries that leave the origin fixed are called Lorentz transformations and give rise to the symmetry known as Lorentz covariance.
• Projective symmetries: These are spatio-temporal symmetries which preserve the geodesic structure of spacetime. They may be defined on any smooth manifold, but find many applications in the study of exact solutions in general relativity.
• Inversion transformations: These are spatio-temporal symmetries which generalise Poincaré transformations to include other conformal one-to-one transformations on the space-time coordinates. Lengths are not invariant under inversion transformations but there is a cross-ratio on four points that is invariant.

Mathematically, spacetime symmetries are usually described by smooth vector fields on a smooth manifold. The underlying local diffeomorphisms associated with the vector fields correspond more directly to the physical symmetries, but the vector fields themselves are more often used when classifying the symmetries of the physical system.

Some of the most important vector fields are Killing vector fields which are those spacetime symmetries that preserve the underlying metric structure of a manifold. In rough terms, Killing vector fields preserve the distance between any two points of the manifold and often go by the name of isometries.

Discrete symmetries

A discrete symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square's original appearance. Discrete symmetries sometimes involve some type of 'swapping', these swaps usually being called reflections or interchanges.

• Time reversal: Many laws of physics describe real phenomena when the direction of time is reversed. Mathematically, this is represented by the transformation, ${\displaystyle t\,\rightarrow -t}$. For example, Newton's second law of motion still holds if, in the equation ${\displaystyle F\,=m{\ddot {r}}}$, ${\displaystyle t}$ is replaced by ${\displaystyle -t}$. This may be illustrated by recording the motion of an object thrown up vertically (neglecting air resistance) and then playing it back. The object will follow the same parabolic trajectory through the air, whether the recording is played normally or in reverse. Thus, position is symmetric with respect to the instant that the object is at its maximum height.
• Spatial inversion: These are represented by transformations of the form ${\displaystyle {\vec {r}}\,\rightarrow -{\vec {r}}}$ and indicate an invariance property of a system when the coordinates are 'inverted'. Said another way, these are symmetries between a certain object and its mirror image.
• Glide reflection: These are represented by a composition of a translation and a reflection. These symmetries occur in some crystals and in some planar symmetries, known as wallpaper symmetries.

C, P, and T symmetries

The Standard model of particle physics has three related natural near-symmetries. These state that the universe in which we live should be indistinguishable from one where a certain type of change is introduced.

• C-symmetry (charge symmetry), a universe where every particle is replaced with its antiparticle
• P-symmetry (parity symmetry), a universe where everything is mirrored along the three physical axes

• T-symmetry (time reversal symmetry), a universe where the direction of time is reversed. T-symmetry is counterintuitive (surely the future and the past are not symmetrical) but explained by the fact that the Standard model describes local properties, not global ones like entropy. To properly reverse the direction of time, one would have to put the big bang and the resulting low-entropy state in the "future." Since we perceive the "past" ("future") as having lower (higher) entropy than the present (see perception of time), the inhabitants of this hypothetical time-reversed universe would perceive the future in the same way as we perceive the past.

These symmetries are near-symmetries because each is broken in the present-day universe. However, the Standard Model predicts that the combination of the three (that is, the simultaneous application of all three transformations) must be a symmetry, called CPT symmetry. CP violation, the violation of the combination of C- and P-symmetry, is necessary for the presence of significant amounts of baryonic matter in the universe. CP violation is a fruitful area of current research in particle physics.

Supersymmetry

A type of symmetry known as supersymmetry has been used to try to make theoretical advances in the standard model. Supersymmetry is based on the idea that there is another physical symmetry beyond those already developed in the standard model, specifically a symmetry between bosons and fermions. Supersymmetry asserts that each type of boson has, as a supersymmetric partner, a fermion, called a superpartner, and vice versa. Supersymmetry has not yet been experimentally verified: no known particle has the correct properties to be a superpartner of any other known particle. Currently LHC is preparing for a run which tests supersymmetry.

Mathematics of physical symmetry

The transformations describing physical symmetries typically form a mathematical group. Group theory is an important area of mathematics for physicists.
Continuous symmetries are specified mathematically by continuous groups (called Lie groups). Many physical symmetries are isometries and are specified by symmetry groups. Sometimes this term is used for more general types of symmetries. The set of all proper rotations (about any angle) through any axis of a sphere form a Lie group called the special orthogonal group ${\displaystyle \,SO(3)}$. (The 3 refers to the three-dimensional space of an ordinary sphere.) Thus, the symmetry group of the sphere with proper rotations is ${\displaystyle \,SO(3)}$. Any rotation preserves distances on the surface of the ball. The set of all Lorentz transformations form a group called the Lorentz group (this may be generalised to the Poincaré group).

Discrete symmetries are described by discrete groups. For example, the symmetries of an equilateral triangle are described by the symmetric group ${\displaystyle \,S_{3}}$.

An important type of physical theory based on local symmetries is called a gauge theory and the symmetries natural to such a theory are called gauge symmetries. Gauge symmetries in the Standard model, used to describe three of the fundamental interactions, are based on the SU(3) × SU(2) × U(1) group. (Roughly speaking, the symmetries of the SU(3) group describe the strong force, the SU(2) group describes the weak interaction and the U(1) group describes the electromagnetic force.)

Also, the reduction by symmetry of the energy functional under the action by a group and spontaneous symmetry breaking of transformations of symmetric groups appear to elucidate topics in particle physics (for example, the unification of electromagnetism and the weak force in physical cosmology).

Conservation laws and symmetry

The symmetry properties of a physical system are intimately related to the conservation laws characterizing that system. Noether's theorem gives a precise description of this relation. The theorem states that each continuous symmetry of a physical system implies that some physical property of that system is conserved. Conversely, each conserved quantity has a corresponding symmetry. For example, the isometry of space gives rise to conservation of (linear) momentum, and isometry of time gives rise to conservation of energy.
The following table summarizes some fundamental symmetries and the associated conserved quantity.

Class	Invariance	Conserved quantity
Proper orthochronous Lorentz symmetry	translation in time   (homogeneity)	energy
	translation in space   (homogeneity)	linear momentum
	rotation in space   (isotropy)	angular momentum
Discrete symmetry	P, coordinate inversion	spatial parity
	C, charge conjugation	charge parity
	T, time reversal	time parity
	CPT	product of parities
Internal symmetry (independent of spacetime coordinates)	U(1) gauge transformation	electric charge
	U(1) gauge transformation	lepton generation number
	U(1) gauge transformation	hypercharge
	U(1)Y gauge transformation	weak hypercharge
	U(2) [ U(1) × SU(2) ]	electroweak force
	SU(2) gauge transformation	isospin
	SU(2)L gauge transformation	weak isospin
	P × SU(2)	G-parity
	SU(3) "winding number"	baryon number
	SU(3) gauge transformation	quark color
	SU(3) (approximate)	quark flavor
	S(U(2) × U(3)) [ U(1) × SU(2) × SU(3) ]	Standard Model

Mathematics

Continuous symmetries in physics preserve transformations. One can specify a symmetry by showing how a very small transformation affects various particle fields. The commutator of two of these infinitesimal transformations are equivalent to a third infinitesimal transformation of the same kind hence they form a Lie algebra.

A general coordinate transformation (also known as a diffeomorphism) has the infinitesimal effect on a scalar, spinor and vector field for example:

${\displaystyle \delta \phi (x)=h^{\mu }(x)\partial _{\mu }\phi (x)}$
${\displaystyle \delta \psi ^{\alpha }(x)=h^{\mu }(x)\partial _{\mu }\psi ^{\alpha }(x)+\partial _{\mu }h_{\nu }(x)\sigma _{\mu \nu }^{\alpha \beta }\psi ^{\beta }(x)}$
${\displaystyle \delta A_{\mu }(x)=h^{\nu }(x)\partial _{\nu }A_{\mu }(x)+A_{\nu }(x)\partial _{\mu }h^{\nu }(x)}$

for a general field, ${\displaystyle h(x)}$. Without gravity only the Poincaré symmetries are preserved which restricts ${\displaystyle h(x)}$ to be of the form:

${\displaystyle h^{\mu }(x)=M^{\mu \nu }x_{\nu }+P^{\mu }}$


where M is an antisymmetric matrix (giving the Lorentz and rotational symmetries) and P is a general vector (giving the translational symmetries). Other symmetries affect multiple fields simultaneously. For example, local gauge transformations apply to both a vector and spinor field:

${\displaystyle \delta \psi ^{\alpha }(x)=\lambda (x).\tau ^{\alpha \beta }\psi ^{\beta }(x)}$
${\displaystyle \delta A_{\mu }(x)=\partial _{\mu }\lambda (x)}$

where ${\displaystyle \tau }$ are generators of a particular Lie group. So far the transformations on the right have only included fields of the same type. Supersymmetries are defined according to how the mix fields of different types.

Another symmetry which is part of some theories of physics and not in others is scale invariance which involve Weyl transformations of the following kind:

${\displaystyle \delta \phi (x)=\Omega (x)\phi (x)}$

If the fields have this symmetry then it can be shown that the field theory is almost certainly conformally invariant also. This means that in the absence of gravity h(x) would restricted to the form:

${\displaystyle h^{\mu }(x)=M^{\mu \nu }x_{\nu }+P^{\mu }+Dx_{\mu }+K^{\mu }|x|^{2}-2K^{\nu }x_{\nu }x_{\mu }}$

with D generating scale transformations and K generating special conformal transformations. For example, N=4 super-Yang-Mills theory has this symmetry while General Relativity doesn't although other theories of gravity such as conformal gravity do. The 'action' of a field theory is an invariant under all the symmetries of the theory. Much of modern theoretical physics is to do with speculating on the various symmetries the Universe may have and finding the invariants to construct field theories as models.

In string theories, since a string can be decomposed into an infinite number of particle fields, the symmetries on the string world sheet is equivalent to special transformations which mix an infinite number of fields.