Generally, the correspondence between continuous symmetries and conservation laws is given by Noether's theorem.
Unitarity is generally required for operators representing transformations of space, time, and spin, since the norm of a state (representing the total probability of finding the particle somewhere with some spin) must be invariant under these transformations.
Written in Dirac notation as standard, the transformations on quantum state vectors are:
There is an additional subtlety that arises in quantum theory, where two vectors that differ by multiplication by a scalar represent the same physical state.
acts on a wavefunction to shift the space coordinates by an infinitesimal displacement Δr.
The exponential functions arise by definition as those limits, due to Euler, and can be understood physically and mathematically as follows.
, acts on a wavefunction to rotate the spatial coordinates of a particle by a constant angle Δθ:
It is not as obvious how to determine the rotational operator compared to space and time translations.
In quantum mechanics, there is another form of rotation which mathematically appears similar to the orbital case, but has different properties, described next.
Spin is a quantity possessed by particles in quantum mechanics without any classical analogue, having the units of angular momentum.
However, unlike orbital angular momentum in which the z-projection quantum number ℓ can only take positive or negative integer values (including zero), the z-projection spin quantum number s can take all positive and negative half-integer values.
This can be used to define a spinor as a column vector of 2s + 1 components which transforms to a rotated coordinate system according to the spin matrix at a fixed point in space.
and is an important quantity for multi-particle systems, especially in nuclear physics and the quantum chemistry of multi-electron atoms and molecules.
The two dimensional quantum harmonic oscillator has the expected conserved quantities of the Hamiltonian and the angular momentum, but has additional hidden conserved quantities of energy level difference and another form of angular momentum.
[5] Lorentz transformations can be parametrized by rapidity φ for a boost in the direction of a three-dimensional unit vector
The term "boost" refers to the relative velocity between two frames, and is not to be conflated with momentum as the generator of translations, as explained below.
and correspondingly, the boost and rotation parameters are collected into another antisymmetric four-dimensional matrix ω, with entries:
The Λ matrices act on any four vector A = (A0, A1, A2, A3) and mix the time-like and the space-like components, according to:
where D(Λ) is a finite-dimensional representation, in other words a (2s + 1)×(2s + 1) dimensional square matrix, and ψ is thought of as a column vector containing components with the (2s + 1) allowed values of σ:
The matrices satisfying the above commutation relations are the same as for spins a and b have components given by multiplying Kronecker delta values with angular momentum matrix elements:
The integers or half-integers m and n numerate all the irreducible representations by, in equivalent notations used by authors: D(m, n) ≡ (m, n) ≡ D(m) ⊗ D(n), which are each [(2m + 1)(2n + 1)]×[(2m + 1)(2n + 1)] square matrices.
Applying this to particles with spin s; In these cases the D refers to any of D(J), D(K), or a full Lorentz transformation D(Λ).
With relativistic quantum mechanics in mind, the time duration and spatial displacement parameters (four in total, one for time and three for space) combine into a spacetime displacement ΔX = (cΔt, −Δr), and the energy and momentum operators are inserted in the four-momentum to obtain a four-momentum operator,
a Casimir operator, is the constant spin contribution to the total angular momentum, and there are commutation relations between P and W and between M and W:
[12] The eight Gell-Mann matrices λn (see article for them and the structure constants) are important for quantum chromodynamics.
They originally arose in the theory SU(3) of flavor which is still of practical importance in nuclear physics.
Symmetries have been proposed to the effect that all fermionic particles have bosonic analogues, and vice versa.
But it has been speculated that dark matter is constitutes gravitinos, a spin 3/2 particle with mass, its supersymmetric partner being the graviton.
Semi-integer spin particles change the sign of their wave function upon a 360 degree rotation (see more in spin–statistics theorem).
This in turn results in degeneracy pressure for fermions—the strong resistance of fermions to compression into smaller volume.