# What are fermionic dimensions

In the **Quantum statistics** the behavior of macroscopic systems is investigated using the methods of quantum mechanics. Similar to classical statistical physics, the starting point is the assumption that the system is in an unknown micro-state. Is the system in a state of the Hilbert space, one speaks of a *pure state*. In analogy to the classical ensemble, one looks at the superposition of different states , the so-called *mixed states*. The density operator is used to describe the quantum mechanical system

Existence is important for quantum statistics *identical particles*, these are quantum objects that cannot be distinguished by any measurement, in other words the Hamiltonian of the system is symmetrical in the particle variables, e.g. B. Degree of spatial and spin freedom. The many-body wave functionψ (1,2, ...,*N*) remains invariant when interchanged, every operator *A.* commutes with a permutation*P* the particle:

Since every permutation of transpositions τ_{ij} can be put together and holds, it makes sense to only consider totally symmetrical or antisymmetrical many-body states:

The experiment shows that nature actually only realizes such states, which can be recognized by the lack of exchange degeneracy; this fact is also called the symmetrization postulate. Particles whose wave function is symmetrical are called bosons, the others are called fermions. In two dimensions there is a phase factor *e*^{iφ} conceivable if interchanged, these particles are called anyons. The spin statistics theorem links the symmetry of the wave function with its spin, fermions have half-integer spin, bosons have integer spin, with anyons other rational relationships can occur. The Fermi-Dirac statistics describe the distribution of fermions over the states of a many-body system, the Bose-Einstein statistics correspondingly for bosons. Examples of quantum statistical effects are (for bosons) the Bose-Einstein condensation, linked with superconductivity (Cooper pairs!) And superfluidity, the cavity radiation of black bodies. For fermions, significant examples of quantum statistical effects are the heat capacity of solids, the band structure of metals and semiconductors, the cohesion of so-called. *white dwarfs* and neutron stars versus self-gravity. In general, the so-called Nernst's theorem (3rd law of thermodynamics*(The absolute zero point can never be reached)*).

### Connection with the rotational behavior of the wave function

Fermions or bosons are known to have *half-integer* or. *integer* Spin (measured in units of , with the quantum of action h). The commutation equation given last is therefore also referred to as the spin statistics theorem. It was proved from general principles of the relativistic quantum field theory by Wolfgang Pauli. That too *Rotational behavior of the wave function* is interesting in this context: For fermions it is true that a spatial rotation of 360^{O} the wave function (Wf) only by 180^{O} changes (Wf -> e^{iπ} Wf = -Wf), while with bosons the wave function is reproduced (Wf -> e^{2iπ} Wf = + Wf). The interchange of two particles can just be achieved through such a 360^{O}-Rotation take place (particle 1 moves to location 2 e.g. on the *upper* Half of a circular line, while particle 2, avoiding a coincidence, on the *lower* Semicircle line moved to the empty location of 1). The result of the above permutation equation fits exactly to the unusual rotational behavior of fermionic wave functions (mathematical structure: see *Double group* SU (2) to the usual turning group SO (3)).

Category: quantum physics

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