Quantum physics applies Hilbert spaces as a realm in which quantum physical modeling is performed. Hilbert spaces are no more and no less than storage media. Storage is achieved in the eigenspaces of operators that map the Hilbert space onto itself. Hilbert spaces can only handle number systems that are division rings. In a division ring all non-zero elements own a unique inverse. Only three suitable division rings exist: real numbers, complex numbers and quaternions. Depending on their dimension these number systems exist in multiple versions that differ in the way that they are ordered.
For separable Hilbert spaces it is possible to construct a category of operators by using the rational elements of a selected number system as the eigenvalues and the members of an orthonormal base of Hilbert vectors as the corresponding eigenvectors. This procedure creates the category of reference operators and their eigenspaces are suitable as parameter spaces. This category can be extended to a category of normal operators that use (almost everywhere) continuous functions in order to define the corresponding eigenvalues that apply the eigenvectors of a selected reference operator. This procedure relates this category of operators to almost everywhere continuous functions.
Every infinite dimensional separable Hilbert space owns a unique companion non-separable Hilbert space. That Hilbert space allows operators to have continuum eigenspaces. Now we define a reference operator that use all elements of a division ring as its eigenvalues and all members of an orthonormal base as the corresponding set of eigenvectors. These sets are no longer countable. We use the same defining function in order to define a corresponding normal operator. This procedure defines a unique companion non-separable Hilbert space.
Now take the differential of the defining function. Where it exists, it defines a new function that can act as the defining function of a new normal operator. That operator does not commute with the previous normal operator.
This is not the only way that a non-commuting operator can be constructed. For example in general quaternions do not commute.
http://vixra.org/abs/1605.0101
Thanks Hans, can we interpret anti commutation as we interpret commutation relations with uncertainity in measurement.
Operators that commute have common eigenfunctions; operators that anticommute are those for which certain quadratic combinations commute, therefore have common eigenfunctions. However the question doesn't make sense because its answer depends on the specific operators involved.
Dear Ujjal,
In a sense, anticommutation is also similarly related to correlation between observables, but with a twist. Here is how I understand the situation:
Roughly speaking, uncorrelated observables correspond to commuting (Hermitian) operators. As you sugggested, and as is apparent from the general Heisenberg-Dirac uncertainty principle, commutator quantifies the inherent uncertainty in the simultaneous measurement of two (correlated) observables.
When dealing with fermionic degrees of freedom, the Pauli-Dirac statistics needs to be built into the analysis by making fermionic fields corresponding to independent degrees to anticommute, instead of the commutation in the bosonic case. However, fermionic fields in themselves are not physical observables. What can be physically measured are generally bosonic currents associated with the fermionic fields, which are constructed from pairs of fermionic operators. If such bosonic currents correspond to mutually uncorrelated observables, then the appropriate fermionic fields will anticommute in a way that makes the bosonic currents to commute with one another.
@ Ujjal
Physical meaning of anticommutation relations?
Anticommutation relations are identical to the commutation relations with the exception that we replace the minus signs with plus sign. This small change in the equations has a dramatic effect on the quantum mechanical behavior of particles in the system.
In particular, it enforces the multi-particle wave functions to be antisymmetric!
In a quaternionic Hilbert space, anti-commutation does not mean that eigenvalues belong to different eigenvectors because parts of the eigenvalues can anti-commutate. The uncertainty relation is due to the fact that the defining functions of the concerned operators are each others Fourier transform. If a distribution owns a Fourier transform, then it means that the distribution owns a displacement generator. That displacement generator corresponds to a differential operator.
The fact that the original distribution is a probability density distribution is a very special situation. It occurs if the distribution corresponds to a coherent location swarm that is characterized by a mostly continuous location density distribution/function. This may happen when the swarm represents the landing locations of a hopping path. This situation is sketched in the link below.
Physical theories can describe these situations by applying the Hilbert space technology. However, physical theories have no explanation for the coherence of the swarms. They just ignore what makes the wave function a coherent descriptor of elementary particles.
http://vixra.org/abs/1606.0028
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