Symmetry in physics

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Symmetry in physics refers to various features of a physical system that can be said to exhibit the property of symmetry—that is, under certain transformations, aspects of these systems are shown to or appear to "be unchanged," according to a particular observation. A symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is "preserved" under some change.

These transformations may be continuous (such as rotations) or discrete (such as reflections) and lead to corresponding types of symmetries. These symmetries are usually formulated mathematically and can be exploited to simplify many problems.

Contents 1 Symmetry as invariance 1.1 Invariance in force 2 Local and global symmetries 3 Continuous symmetries 3.1 Spacetime symmetries 4 Discrete symmetries 4.1 C, P, and T symmetries 4.2 Supersymmetry 5 Mathematics of physical symmetry 6 Applications of symmetry 6.1 Conservation laws and Noether's theorem 7 See also 8 References 9 External links

[edit] Symmetry as invariance

Invariance is usually specified mathematically by transformations that leave some quantity unchanged. This idea can apply to basic real-world observations. For example, the temperature in a room may be constant. The temperature being independent of position within the room, it is said that the temperature is "unchanged" by a shift in the measurer's position.

Similarly, an unmarked ping-pong ball, when rotated about its centre, will appear exactly as it did before the rotation. The ping-pong ball is said to exhibit spherical symmetry. A rotation about any axis of the ball will preserve how the ball "looks."

[edit] Invariance in force

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

For example, an electrical wire is said to exhibit cylindrical symmetry, because the electric field strength at a given distance r₀ from an electrically charged wire of infinite length will have the same magnitude at each point on the surface of a cylinder (whose axis is the wire) with radius r₀. Hence rotating the wire about its own axis does not change its position, hence it will preserve the field. The field strength at a rotated position is the same, but its direction is rotated accordingly. These two properties are interconnected through the more general property that rotating any system of charges causes a corresponding rotation of the electric field.

In Newton's theory of mechanics, given two equal masses m starting from rest 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 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.

[edit] Local and global symmetries

Main articles: Global symmetry and Local symmetry

Symmetries may be broadly classified as global and local. A global symmetry is one that holds at all points of spacetime, whereas a local symmetry is one that only holds on a certain subset of the whole spacetime. Local symmetries tend to play an important role in physics, as measurements are performed in a limited region of space (or spacetime).

[edit] Continuous symmetries

The two examples of rotational symmetry described above - spherical and cylindrical - are each instances of continuous symmetry. These are characterised by 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 magnitude will be the same on any given cylinder. Mathematically, continuous symmetries are usually described by continuous or smooth functions. An important subclass of continuous symmetries in physics are spacetime symmetries.

[edit] Spacetime symmetries

Main article: Spacetime symmetries

Spacetime symmetries are those continuous symmetries that involve transformations of space and time. These may be further divided into 3 categories. Many symmetries in physics are described by continuous changes of the spatial geometry associated with a physical system (' spatial symmetries '), others only involve continuous changes in time (' temporal symmetries ') or continuous changes in both space and time (' spatio-temporal symmetries ').

• Time translation: A physical system may have the same features over a certain interval of time δt ; this is expressed mathematically as invariance under the transformation for any real numbers t and t + a in the interval. For example, in classical mechanics, a particle solely acted upon by gravity will have gravitational potential energy when suspended from a height 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) t₀ and also at t₀ + 3, say, the particle's total gravitational potential energy will be preserved.

• Spatial translation: These spatial symmetries are represented by transformations of the form 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 into two types, namely, proper rotations and improper rotations. The former are just the 'ordinary' rotations; mathematically, they are square matrices with unit determinant. The latter are represented by square matrices with determinant -1 and consist of a proper rotation composed with a spatial reflection (inversion). For example, a ping-pong ball has rotational symmetry where the rotations are proper. 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 1-1 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. The article isometries in physics discusses examples of these symmetries in more detail.

[edit] Discrete symmetries

Main article: Discrete symmetry

A discrete symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete symmetry, as only rotations by integral multiples of 90 degrees will preserve the square's original outlook. Discrete symmetries often 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, . For example, Newton's second law of motion still holds if, in the equation , t is replaced by − t. This may be illustrated by describing the motion of a particle thrown up vertically (neglecting air resistance). For such a particle, position is symmetric with respect to the instant that the object is at its maximum height. Velocity at reversed time is reversed.

• Spatial inversion: These are represented by transformations of the form and indicate an invariance property of a system when the coordinates are 'inverted'. Example 4 above illustrates this spatial symmetry.

• Glide reflection: These are represented by a composition of a translation and a reflection. These symmetries occur in certain crystals.

[edit] C, P, and T symmetries

The Standard model of particle physics has three related natural near-symmetries. These state that the universe is indistinguishable from one where...:

• C-symmetry (Charge symmetry) ... every particle is replaced with its antiparticle.
• P-symmetry (Parity symmetry) ... the universe is reflected as in a mirror.
• T-symmetry (Time symmetry) ... the direction of time is reversed. (This 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 properties like entropy. To properly time-reverse the universe, you would have to put the big bang and the resulting low-entropy conditions in the "future". Since our experience of time is related to entropy, the inhabitants of the resulting universe would then see that as the past.)

Each of these symmetries is broken, but the Standard Model predicts that the combination of the three (that is, the three transformations at the same time) must be a symmetry, known as CPT symmetry. CP violation, the violation of the combination of C and P symmetry, is a currently fruitful area of particle physics research, as well as being necessary for the presence of significant amounts of matter in the universe and thus the existence of life.

[edit] Supersymmetry

Main article: Supersymmetry

Extensions of symmetry to the concept of supersymmetry have been used to try to make theoretical advances in the standard model. Roughly speaking, supersymmetry is based on the idea that there is one remaining physical symmetry beyond those that are well-understood, a symmetry between bosons and fermions, so that each boson would have a symmetry partner fermion, called a superpartner, and vice versa. There are significant unsolved problems with the theory of supersymmetry, including that no known particle has the correct properties to be a superpartner of any other known particle, so that if superpartners exist, they apparently all must have greater mass than existing particle accelerators have been capable of generating.

[edit] Mathematics of physical symmetry

Main article: Symmetry group

Many of the important transformations describing physical symmetries form a group. This has led to group theory being one of the areas of mathematics most studied by 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 . Thus, the symmetry group of the ping-pong ball with proper rotations is . 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 tend to be described by discrete groups. For example, the symmetries of an equilateral triangle are described by the symmetric group .

In the Standard model of particle physics, the gauge group used to describe 3 of the fundamental interactions is SU(3) × SU(2) × U(1). 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).

A physical theory based on groups is sometimes called a Gauge theory and the symmetries natural to such a theory are called Gauge symmetries.

[edit] Applications of symmetry

Physical problems can be simplified by noticing any symmetries that a system possesses.

[edit] Conservation laws and Noether's theorem

Main article: Noether's theorem

The symmetry properties of a physical system are intimately related to the conservation laws characterizing that system. Noether's theorem makes this fact precise. Roughly, that theorem states that each symmetry of a physical system implies that some physical property of that system is conserved, and conversely that each conserved quantity has a corresponding symmetry.

[edit] See also

• Field (physics)
• Gauge theory
• Lagrangian
• Noether's theorem
• Particle physics and representation theory
• Principle of relativity
• Spacetime symmetries
• CPT symmetry

[edit] References

• Brading, K., and Castellani, E., eds., 2003. Symmetries in Physics: Philosophical Reflections. Cambridge Uni. Press.
• Rosen, Joe, 1995. Symmetry in Science: An Introduction to the General Theory. Springer-Verlag.
• Van Fraassen, B. C., 1989. Laws and symmetry. Oxford Uni. Press.

[edit] External links

• Stanford Encyclopedia of Philosophy: Symmetry by Brading and Castellani.