Invariant (physics)

In theoretical physics, an invariant is a special type of observable. An observable is a measurable part of a physical system. An invariant remains unchanged when a transformation is applied to that system. A transformation is a way of changing how we view or measure something. Invariance can also describe physical laws. A law is invariant if its form does not change during a transformation. This means the mathematical equations used to describe the law stay the same. This concept is vital to modern physics. Many scientific theories are built entirely around symmetries and invariants.

Invariants are deeply tied to the symmetries of a system's environment. Symmetry refers to a pattern that stays consistent under certain changes. When a system possesses a continuous symmetry, it leads to a specific result. A mathematician named Emmy Noether proved this connection with Noether's theorem. Her theorem states that any continuous symmetry results in a fundamental conservation law. This means if a physical system has a certain symmetry, a specific quantity must be conserved. Conservation means that the quantity stays constant over time.

We can see these connections in classical and quantum mechanics. For example, space has a symmetry called translation. Translation in space means moving from one location to another. Because space is invariant under translation, momentum is an invariant. This results in the conservation of momentum. Time also has a similar symmetry called translation in time. Because the origin of time is invariant, energy is an invariant. This leads directly to the conservation of energy.

Different types of transformations affect different quantities in various ways. Some quantities are invariant under some transformations but not others. The velocity of a particle is one such example. Velocity is invariant when you switch between rectangular and curvilinear coordinate systems. However, velocity is not invariant when you transform between different frames of reference. A frame of reference is a way of measuring motion. In contrast, some quantities are always invariant regardless of the change. The speed of light is a constant that does not change. The mass and charge of a particle are also always invariant.

In special relativity, we look at spacetime Lorentz transformations. These involve two reference frames moving at different speeds relative to each other. Even under these complex changes, the speed of light remains invariant. We also use Galilean transformations to study frames moving at low velocities. Under a Galilean transformation, time and acceleration remain invariant. In crystals, we see a different kind of symmetry. The electron density in a crystal is periodic. This means it is invariant with respect to discrete translations by unit cell vectors. In very few materials, this symmetry can actually be broken.

Physicists also discuss how equations behave under transformations. A law is considered invariant if its predictions remain unchanged. This usually means the type of differential equations used stays the same. The Klein–Gordon equation and the Dirac equation are examples of invariant equations. They keep their written form under the coordinate transformations of special relativity. The Schrödinger equation is different in this regard. It does not keep its written form under those same transformations. Therefore, the Schrödinger equation is not invariant under special relativity.

There is often informal usage of specific terms in physics labs. Many physicists use the word "covariant" as a synonym for "invariant." For example, they might say the Schrödinger equation is not covariant. However, this is not strictly accurate. It is more precise to say the Klein–Gordon and Dirac equations are invariant. To avoid confusion, scientists should indicate which transformation they are evaluating. The mathematical properties of invariance are further generalized in tensor mathematics. These properties are known as covariance and contravariance. These concepts are used frequently in electromagnetism and relativity.