# Symmetry laws

## Symmetry laws (physics)

The physical laws which are expressions of symmetries. The term symmetry, as it is used in mathematics and the exact sciences, refers to a special property of bodies or of physical laws, namely that they are left unchanged by transformations which, in general, might have changed them. For example, the geometric form of a sphere is not changed by any rotation of the sphere around its center, and so a sphere can be said to be symmetric under rotations. Symmetry can be very powerful in constraining form. Indeed, referring to the same example, the only sort of surface which is symmetric under arbitrary rotations is a sphere.

The concept that physical laws exhibit symmetry is more subtle. A naive formulation would be that a physical law exhibits symmetry if there is some transformation of the universe that might have changed the form of the law but in reality does not. However, the comparison of different universes is generally not feasible or desirable. A more fruitful definition of the symmetry of physical law exploits locality, the principle that the behavior of a given system is only slightly affected by the behavior of other bodies far removed from it in space or time. Because of locality, it is possible to define symmetry by using transformations that do not involve the universe as a whole but only a suitably isolated portion of it. Thus the statement that the laws of physics are symmetric under rotations means that (say) astronauts in space would not be able to orient themselves—to determine a preferred direction—by experiments internal to their space station. They could do this only by referring to weak effects from distant objects, such as the light of distant stars or the small residual gravity of Earth.

#### Symmetries of space and time

Perhaps the most basic and profound symmetries of physical laws are symmetry under translation in time and under translations in space.

The statement that fundamental physical laws are symmetric under translation in time is equivalent to the statement that these laws do not change or evolve. Time-translation symmetry is supposed to apply, fundamentally, to simple isolated systems. Large complicated systems, and in particular the universe as a whole, do of course age and evolve. Thus in constructing the big-bang model of cosmology, it is assumed that the properties of individual electrons or protons do not change in time, although of course the state of the universe as a whole, according to the model, has changed quite drastically.

The statement that fundamental laws are symmetric under translations in space is another way of formulating the homogeneity of space. It is the statement that the laws are the same throughout the universe. It says that the astronauts in the previous example cannot infer their location by local experiments within their space station. The power of this symmetry is that it makes it possible to infer, from observations in laboratories on Earth, the behavior of matter anywhere in the universe.

The symmetry of physical law under rotations, mentioned above, embodies the isotropy of space.

In the mathematical formulation of dynamics, there is an intimate connection between symmetries and conservation laws. Symmetry under time translation implies conservation of energy; symmetry under spatial translations implies conservation of momentum; and symmetry under rotation implies conservation of angular momentum. *See* Angular momentum, Conservation of energy, Conservation of momentum

The fundamental postulate of the special theory of relativity, that the laws of physics take the same form for observers moving with respect to one another at a fixed velocity, is clearly another statement about the symmetry of physical law. The idea that physical laws should be unchanged by such transformations was discussed by Galileo, who illustrated it by an observer's inability to infer motion while on a calm sea voyage in an enclosed cabin. The novelty of Einstein's theory arises from combining this velocity symmetry with a second postulate, deduced from experiments, that the speed of light is a universal constant and must take the same value for both stationary and uniformly moving observers. *See* Galilean transformations, Relativity

#### Discrete symmetries

Before 1956, it was believed that all physical laws obeyed an additional set of fundamental symmetries, denoted *P*, *C*, and *T*, for parity, charge conjugation, and time reversal, respectively. Experiments involving particles known as *K* mesons led to the suggestion that *P* might be violated in the weak interactions, and violations were indeed observed. This discovery led to questioning—and in some cases overthrow—of other cherished symmetry principles.

Parity, *P*, roughly speaking, transforms objects into the shapes of their mirror images. If *P* were a symmetry, the apparent behavior of the images of objects reflected in a mirror would also be the actual behavior of corresponding real objects. *See* Parity (quantum mechanics)

Charge conjugation, *C*, changes particles into their antiparticles. It is a purely internal transformation; that is, it does not involve space and time. If the laws of physics were symmetric under charge conjugation, the result of an experiment involving antiparticles could be inferred from the corresponding experiment involving particles.

Remarkably, by combining the transformations *P* and *C*, a result is obtained, *CP*, which is much more nearly a valid symmetry than either of its components separately. However, in 1964 it was discovered experimentally that even *CP* is not quite a valid symmetry.

Although the preceding discussion has emphasized the failure of *P*, *C*, and *CP* to be precise symmetries of physical law, both the strong force responsible for nuclear structure and reactions and the electromagnetic force responsible for atomic structure and chemistry do obey these symmetries. Only the weak force, responsible for beta radioactivity and some relatively slow decays of exotic elementary particles, violates them. Thus these symmetries, while approximate, are quite useful and powerful in nuclear and atomic physics. *See* Electroweak interaction, Fundamental interactions, Weak nuclear interactions

The operation of time-reversal symmetry, *T*, involves changing the direction of motion of all particles. For example, it relates reactions of the type A + B → C + D to their reverse C + D → A + B. No direct violation of *T* has been detected.

Time-reversal symmetry, even if valid, applies in a straightforward way only to elementary processes. It does not, for example, contradict the one-way character of the second law of thermodynamics, which states that entropy can only increase with time. *See* Thermodynamic principles, Time, arrow of

Fundamental principles of quantum field theory suggest that the combined operation *PCT*, which involves simultaneously reflecting space, changing particles into antiparticles, and reversing the direction of time, must be a symmetry of physical law. Existing evidence is consistent with this prediction. *See* CPT theorem

#### Internal symmetry

Internal symmetries, like *C*, do not involve transformations in space-time but change one type of particle into another. An important, although approximate, symmetry of this kind is isospin or i-spin symmetry. It is observed experimentally that the strong interactions of the proton and neutron are essentially the same. *See* I-spin

There have been several successful predictions of the existence and properties of new particles, based on postulates of internal symmetries. Perhaps the most notable was the prediction of the mass and properties of the &OHgr;^{-} baryon, based on an extension of the symmetry group SU(2) of isospin to a larger approximate SU(3) symmetry acting on strange particles as well. These symmetries were an important hint that the fundamental strong interactions are at some level universal, that is, act on all quarks in the same way, and thus paved the way toward modern quantum chromodynamics, which does implement such universality. *See* Baryon, Quantum chromodynamics, Unitary symmetry

Much simpler mathematically than SU(2) internal symmetry, but quite profound physically, is U(1) internal symmetry. The important case of the electric charge quantum number will be considered. The action of the U(1) internal symmetry transformation with parameter λ is to multiply the wave function of a state of electric charge *q* by the factor *e*^{iλq}. An amplitude between two states with electric charges *q _{r}* and

*q*will therefore be multiplied by a factor

_{s}*e*

^{iλ(qs-qr)}. Since the physical predictions of quantum mechanics depend on such amplitudes, these predictions will be unchanged only if the phase factors multiplying all nonvanishing amplitudes are trivial. This will be true, in turn, only if the amplitudes between states of unequal charge vanish; that is, if charge-changing amplitudes are forbidden, which is just a backhanded way of expressing the conservation of charge.

*See*Quantum mechanics, Quantum numbers

#### Localization of symmetry

The concept of local gauge invariance, which is central to the standard model of fundamental particle interactions, and in a slightly different form to general relativity, may be approached as a generalization of the U(1) internal symmetry transformation, where a parameter λ independent of space and time appears. Such a parameter goes against the spirit of locality, according to which each point in space-time has a certain independence. There is therefore reason to consider a more general symmetry, involving a space-time-dependent transformation in which the wave function is multiplied by *e*^{iλ(x,t)q(x,t)}, where *q*(*x,t*) is the density of charge at the space-time point (*x,t*). These transformations are much more general than those discussed above, and invariance under them leads to much more powerful and specific consequences.

For electromagnetism, the required interactions of matter with the electromagnetic field are predicted precisely. Thus, the theory of the electromagnetic field—Maxwell's equations and quantum electrodynamics—can be said to be the unique ideal embodiment of the abstract concept of a space-time-dependent symmetry, that is, of local gauge symmetry. *See* Maxwell's equations, Quantum electrodynamics