In quantum mechanics, symmetries play a fundamental role in describing and understanding physical systems. A group acting as symmetries in a quantum system refers to a mathematical group whose elements represent transformations or operations that leave the system invariant or unchanged. These transformations preserve the underlying physics and properties of the system.
Let's consider a quantum system described by a Hilbert space, which represents the possible states of the system. A group acting as symmetries in this context can be understood as a set of unitary operators that act on the states of the system.
Mathematically, a group is defined as a set of elements along with a binary operation (usually denoted as *) that satisfies the following properties:
Closure: If two elements from the group are combined using the binary operation *, the result is still an element of the group.
Identity: There exists an identity element (often denoted as e) in the group such that for any element g in the group, g * e = e * g = g.
Inverse: For every element g in the group, there exists an inverse element (denoted as g^(-1)) such that g * g^(-1) = g^(-1) * g = e.
Associativity: The binary operation * is associative, meaning that (g1 * g2) * g3 = g1 * (g2 * g3) for any three elements g1, g2, and g3 in the group.
When a group acts as symmetries in a quantum system, it means that each element of the group corresponds to a transformation that leaves the system's Hamiltonian (the operator representing the energy of the system) invariant. This implies that the transformation does not change the physical observables or the probabilities associated with different measurement outcomes.
The representation of a group acting on a quantum system can be expressed through a unitary operator U, where U(g) represents the transformation corresponding to the group element g. The action of the group on a quantum state |ψ⟩ is given by U(g)|ψ⟩.
By studying the symmetries of a quantum system, one can gain insight into various properties, conservation laws, and degeneracies associated with the system. Symmetries also play a crucial role in classifying and understanding different types of quantum systems and their behavior.
Note that different groups can act as symmetries in different quantum systems, depending on the specific physical properties and symmetries of the system under consideration. Examples of such groups include the unitary group, special unitary group, orthogonal group, and rotation group, among others.