Thus, once we settle on any unitary representation of the gammas, it is final provided we transform the spinor according to the unitary transformation that corresponds to the given Lorentz transformation.
32.
The density matrix of a KMS state is related to unitary transformations involving time translations ( or time translations and an internal symmetry transformation for nonzero chemical potentials ) via the Tomita Takesaki theory.
33.
If " T " is a non-negative operator on a finite-dimensional Hilbert space, then all square roots of " T " are related by unitary transformations.
34.
Remark : Some intuition for the mean ergodic theorem can be developed by considering the case where complex numbers of unit length are regarded as unitary transformations on the complex plane ( by left multiplication ).
35.
Therefore, any operator " O " in the Dirac-Pauli representation upon which we perform a bi-unitary transformation, will be given, for an " at rest " fermion, by:
36.
To see why the \ tilde { \ Omega } terms are called ` counter-rotating'consider a unitary transformation to the interaction or Dirac picture where the transformed Hamiltonian H _ { 1, I } is given by
37.
In mathematics, the "'Weil Brezin map "', named after Andr?Weil and Jonathan Brezin, is a unitary transformation that maps a Schwartz function on the real line to a smooth function on the Heisenberg manifold.
38.
As one example, when an eigenproblem corresponding to some system ( whether quantum or classical ) has a physical symmetry ( e . g . it is rotationally invariant ), this means that the unitary transformation operator corresponding to the symmetry commutes with the eigenproblem.
39.
The problem can be analyzed more easily by moving into the interaction picture, defined by the unitary transformation \ tilde { M } = UMU ^ \ dagger, where M is an arbitrary operator, and U = e ^ { i ( H _ S + H _ B ) t }.
40.
This is a projection, in the sense that the Euclidean distance to the constraint is minimized, because ( i ) the discrete Fourier transform, as a unitary transformation, preserves distance, and ( ii ) rescaling the modulus ( without modifying the phase ) is the smallest change that realizes the modulus constraint.
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