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Classical Description Quantization Remember Reverse approach

About Anyons (not Anions !)

March 29, 2016

Classical Description Quantization Remember Reverse approach

Outline

(^1) Historical background

(^2) Classical Description

(^3) Quantization

(^4) Remember

(^5) Reverse approach

Classical Description Quantization Remember Reverse approach

When and Why.

Students in physics are always taught that : because of indistinguishability (postulate) wave function can be either symmetric or antisymmetric (postulate) ...with respect to the interchange of two particle coordinates (creation of the mind) Can we find a rigorous statement or proof to reformulate this?

Classical Description Quantization Remember Reverse approach

When and Why.

Where is the problem exaclty? introduction of particle indices, ok .... but it brings elements of nonobservable character! the word interchange has no physical meaning

Classical Description Quantization Remember Reverse approach

Impact of Indistinguishability

Impact of indistinguishability is not only a characteristic of quantum mechanics, it can be seen in classical statistical mechanics too Gibb’s paradox computation of zero-entropy change correction of the volume in phase space

Classical Description Quantization Remember Reverse approach

Impact of Indistinguishability

If we have N particules, let SN be the discrete (finite) group of permutation of coordinates on points in the cartesian product X N the real new physical phase space is the quotient X N^ /SN locally isomorphic to X N^ outside of its singular points X N^ /SN^ has singular points

Classical Description Quantization Remember Reverse approach

Topology

It can be shown that X N^ /SN^ = EN^ × r (n, N) where r (n, N) = EnN−n/SN^. The important result being that the structure of r (n, N) depends on the dimension n and if the singular points are included or not. with the singular point : simply connected without singular points, n = 1 , 2, infinitiely connected without singular points, n > 2, doubly connected the doubly connectedness property will give birth to the notion of anti- and symmetric constraint

Classical Description Quantization Remember Reverse approach

Some Practical Considerations

Suppose 2 electrons separated by a large distance, why should we impose a correlation between them by imposing the symmetric or antisymmetric on their wave function ?! in the new formulation, no need for a symmetrisation postulate the configuration space of two identical particules is locally isomorphic to that of two nonidentical particules except at the singularity (when the particules meet=when they are not separated by a large distance) the singularity of the topology and its consequences appear only when the particles collide (are close one to another)

Classical Description Quantization Remember Reverse approach

Quantization in 1D

On the singular boundary we need to impose something. particle constrained in a domain → 0 on ∂ → we recover the fermions system vanishing normal derivative on ∂ → we recover the bosons system these are too strong requirements instead local conservation of probability on ∂ ψ∗(x, 0 ) (^) ∂∂z ψ(x, 0 ) − (^) ∂∂z ψ∗(x, 0 )ψ(x, 0 ) = 0 ∂ ∂z ψ(x,^0 ) =^ ηψ(x,^0 ) η = 0, bosons, η = ∞, fermions!

Classical Description Quantization Remember Reverse approach

Quantization in 2D, 3D

With the same technique we arrive at the natural condition on the singular boundary

ψ(r , φ + 2 π) = exp[iζ]ψ(r , φ)

but in 3D a specific property of the projective spaces will imply that only ζ = 0 , π is possible (0 for bosons, π for fermions), but no contradictions at all on ζ in 2D, it can be chosen freely.

Classical Description Quantization Remember Reverse approach

Unified Manifold Description

Can you describe the phase space with one theory and one manifold such that

Classical Description Quantization Remember Reverse approach

Unified Manifold Description

In 3D, regardless of the distance d between the particules (small or large), the intrinsic correlation is introduced the same way with only two choices : symmetric → bosons antisymmetric → fermions

Classical Description Quantization Remember Reverse approach

Contradiction?

2 D and 1D systems are just specific 3D systems, so when you allow your real quantum system to evolve with dimension constraints : it changes its very fundamental nature and properties at the highest description level We expect extreme changes and specials results from : fullerenes graphenes ...

Classical Description Quantization Remember Reverse approach

What are we looking for?

Do we know in advance what we are looking for? Is the existence of anyons stated beforehand in the empiric approach ? the very observation of a fractional quantized Hall conductivity, a vanishing longitudinal conductivity and a mobility gap between the ground state and the first excited states, when combined with general principles of physics, will force us to accept the notion of Anyons a two dimensional system that shows the fractional quantum Hall effect must have quasi-particles that carry a fraction of an electron charge