Simultaneous Projection of Products of Quantum Fields

On a formal level we derive these products of noncommutative field operators if we project the usual product of commutative fields. So let Pη be the projection of a single field as defined in the previous section, letW(ψ₁, . . . , ψ_n) be a product of field operators on a suitable (Fermionic) Fock space. The mapPⁿ_η defined by

Pⁿ_η(W) :=hP_η⊗. . .⊗P_η, ψ₁⊗. . .⊗ψ_ni

maps simultaneously all n-fields in the lowest Landau level on a quasi two-plus-one dimen-sional system.

Pⁿ_η(W) =ψ₁^P ∗ψ₂^P ∗ · · · ∗ψ_n^P

withψ_i^P ≡ψ(η), defined in the previous section and the star denotes the product on the field algebra of the projected fields, which is related to the Wigner Moyal star product (⋆) on the noncommutative plane. For example we may apply this map to a Lagrangian term:

P³_η( ¯ψ⊗γ^µa_µ⊗ψ) = ( ¯ψ^P ⋆(γ^µa^P_µ)⋆ ψ^P)(η).

instead of the map hI⊗I⊗I,ψ¯⊗γ^µa_µ⊗ψi. The meaning of a^P_µ ≡a_µ(η) we will discuss later on.

Projection of a Coulomb Potential

The problem which arises in context with the Coulomb potential is, that it is not apriori Fourier transformable. However this problem can be circumvented by the use of a suitable limit of a Yukawa like potential together with the theorem of B. Levi on the monotonic con-vergence, the theorem on the majorized convergence by Lebesgue also holds. The projection is applied to a special representation of the Coulomb potential in terms of a Yukawa poten-tial, which is Fourier transformable. So letV(x−y) be the usual Coulomb potential in three dimensions and let Y_α(x−y) be a Yukawa potential with dumping parameter α > 0. The Coulomb potential expressed in terms of the Yukawa potential is given by

V(x−y) = lim

α→0Y_α(x−y) = lim

α→0

−g² e^−α|x−y|

|x−y| .

Where g² = e²/(4πǫ), x,y ∈R³ and let ˇY_α(k), k∈ R³ be the Fourier-transformed Yukawa potential:

Yˇ_α(k) =−4πg² 1 k²+α². The projected Coulomb potential is given by

V^P(η₁−η₂) =hPη1⊗Pη2, V(x−y)i and is expressed through

V^P(η₁−η₂) = lim

α→0

1 (2π)³

Z

d³kYˇ_α(k) e^−12(liki)2 e^{−ikη1+ikη2} (4.4.7) since [η₁ⁱ, η₂^j] = 0 and w.l.o.g. η₃= 0. At this point it should be mentioned that the classical Limit x → y does not make sense for the unbounded selfadjoint operators η_i^j. A suitable (or pragmatically) treatment is to evaluate the difference η₁−η₂ in appropriate states. It means that we can minimize the distance but it will never go to zero. Therefore the Pauli exclusion principle is always fulfilled. There exists automatically a minimal distance and this is where the difference is evaluated in coherent states, for example in the ground state of a harmonic oscillator. In this case it is clear that these states are those where we should obtain an incompressible (Fermi) liquid.

If we evaluate the differences of the non-commuting coordinates in a (coherent) stateω₀, then we obtain the minimal distance by evaluating

ω0(V^P(η₁−η₂)) = lim

α→0

1 (2π)³

Z

d³kYˇα(k)e^−12(liki)2 ω0

e^−ikη1e^ikη2

= lim

α→0

1 (2π)³

Z

d³kYˇ_α(k)e^{−l2B(k21+k22)+14l2zk32}. (4.4.8) The minimal distance rmin can be defined by the integral and depends on the evaluation in coherent states:

1

r_min := lim

α→0

4π (2π)³

Z

d³k e^−12(liki)2 k²+α² ω0

e^−ikη1e^ikη2

. (4.4.9)

This integral is finite at least if we chose states from Schwartz space the coherent states respectively. The Schwartz function regularizes the integral for large momenta and since α >0 it is well defined on all onR³.

The projection of the Coulomb potential leads to an intrinsic quantised distance between the particles for fixed magnetic fields since it depends on the magnetic length. The simplest example for a minimal distance we calculate forl_z = 4l_B and obtain

ω₀(V^P(η₁−η₂)) = −2g

√π

l_B (4.4.10)

thus r_min = l_B/(2√

π). The third component l_z may be chosen different depending on the confinement potential. It may be reduced such that it gives just a neglectable contribution.

We mapped the Coulomb potential in three dimensions by freezing out the third compo-nent and the radial contribution in coherent states to a screened Coulomb potential in two non-commutative dimensions, which leads in the minimal distance regime to a constant factor and thus to an incompressible liquid. A related discussion on the evaluation of the Coulomb potential can be found as a remark in [BvS94], however they consider the potential only in two dimensions. We stress that the Coulomb potential and the Yukawa potential respectively differs in two dimensions from that in tree dimensions and the evaluation in states leads then to different results which can be explained for instance with the integration measure.

Projection of the Charge Density

The projection of the charge density can easily be performed in our formalism. It is just the projection of the product of the spinorsψ⁺(x)ψ(x) so we obtain

ρ^P(x⁰,η) := P_η²(ψ⁺(x)⊗ψ(x))

= ((ψ⁺)^P ⋆ ψ^P)(x⁰,η)

and the Coulomb interaction term of two particles would result in

ρ(x)V(x−y)ρ(y)7→ρ^P(x⁰₁,η₁)⋆ V^P(η₁−η₂)⋆ ρ^P(x⁰₂,η₂) , x⁰₁ ≡x⁰₂. Projection of an Action

Some more questions arise in this context, first how can we formulate normal-ordering and then how can we formulate a propagator of the projected fields or an action principle. It is the problem which also arises in a field theory on the noncommutative Minkowski space but with a slight modification due to the different commutation relation between the (quantum) coordinates. However since we have a special realization we may use the same concepts.

The QED action given in the previous section has to be projected to the Hall system. The problem which arises here is how to interpret the integration measure. Here the methods of measure theory should clarify what we have to do. The usual measured⁴x is turned into an

operator valued measure, the spectral measure, which we will denote by dE(ψ₁, . . . , ψ_n;Pⁿ_η) and the integration should be considered as a positive trace in a similar way to [DFR95].

S = Z

O

d⁴x L(ψ₁, . . . , ψ_n)→S^P = Z

dE(ψ₁, . . . , ψ_n;Pⁿ_η)L^P(ψ₁^P, . . . , ψ_n^P).

For example the term where the Fermions ψcouples to a field a_µ is given by S_I^P =

Z dx⁰

Z

dE( ¯ψ, a^P_µ, ψ;P³_η) LI(ψ₁^P, a^P_µ, ψ^P_n)

= Z

dx⁰dη₁dη₂ (ψ₁^P ⋆ γ^µa^P_µ ⋆ ψ^P_n)(x⁰,η)

= Z

dx⁰ tr_η[(ψ^P₁ ⋆ γ^µa^P_µ ⋆ ψ^P_n)(x⁰,η)].

The trace trηhas to be well defined or in other words it has to be a positive linear functional.

Projection of the Yang Mills and Chern Simons Term

If we take the definition of the fluctuating U(1) Chern Simons gauge field a_µ from section 4.1 in four or in three dimensions respectively and if we assume that there exists the Fourier transformed

aµ(x) = 1 (2π)⁴

Z

d⁴k aµ(k) e^ikx and their derivative fields

a_µ,ν(x) =∂_νa_µ= 1 (2π)⁴

Z

d⁴k a_µ(k)k_ν e^ikx, then we may define a projection of the field strength

f_µν^P (η) :=Pⁿ_η(f(x)) = Pⁿ_η(f_µν)

= f_µν^P (η)

= (a^P_µ,ν −a^P_ν,µ+a^P_µ ⋆ a^P_ν −a^P_ν ⋆ a^P_µ)(η) with the noncommutative analog of the derivative defined via (4.4.5):

a^P_µ,ν(η) = 1 (2π)⁴

Z

d⁴k a_µ(k) k_ν e^−(lik

i)2 4 e^ikη.

In the following we drop the index “P”, the Yang Mills term is then given in non-local coordinates:

Y M(η) = ¹₄ (f_µν⋆ f^µν)(η) dη₀dη₁dη₂dη₃. In the same manner the Pontryagin term in non-local coordinates is:

P J(η) = ¹₄ ε^µνρσf_µν⋆ f_ρσ dη₀dη₁dη₂dη₃.

The Chern Simons term on a suitable surface is also projected straight forward and expressed in non-local coordinates we have:

W(η) = (ε^µνρ(a_µ⋆ a_ρ,µ+ ²₃a_µ⋆ a_ν⋆ a_ρ))(η)dη₀dη₁dη₂.

Therefore the Chern Simons theory in a Landau level is a non-commutative field theory the noncommutative Chern Simons theory. Susskind derived a similar form of the Chern Simons term via fluid dynamics, where it is considered that the particles occupy a finite, hard disc to satisfy a granular microscopic picture of a Hall fluid [Sus01].

What we immediately recognize is that we fail if we want to obtain the Chern Simons term via projection from the Pontryagin Lagrangian. So far there is a further difficulty since we have to say what Stokes theorem, Poincar´e’s lemma and (cyclic)-cohomology is in a noncommutative space. Nevertheless we observe

R d⁴x ε^µνρσf_µν(x)f_ρσ(x) −−−−−−−−−−−−−→^{(P,4d→3d and LLL)} trε^µνρσfˆ_µν(η)⋆fˆ_ρσ(η)



y^f∧f=dW

 y^?

R d³ε^µνρσ(a_µ∂_νa_σ+²₃a_µa_νa_σ) −−−−−→^(P,LLL) trε^µνσ(ˆa_µ⋆ ∂_νaˆ_σ+²₃ˆa_µ⋆ˆa_ν⋆ˆa_σ) .

The Chern Simons Lagrangian describes a gauge invariant theory modulo Z on a manifold without boundary especially in R³ where all fields vanish at infinity. On manifolds with boundary we may require that all gauge transformations on the boundary vanish. This restricts the underlying gauge group as already mentioned in the previous chapter.

After we project the gauge fields we have a noncommutative version of the Chern Simons Lagrangian. However we may clarify whether we still have at least a noncommutative version of a gauge invariant theory. There arises for instance some problems if we want to describe noncommutative gauge fields. It turns out that there exist noSU(N) noncommutative fields due to a lack of a proper definition of a determinant. From that point of view we would prefer theU(N) gauge fields only. We may quantize the theory in terms of (noncommutative) BRST quantization. However the description of noncommutative gauge theories is even in the classical case still an open question. There are some different approaches to define gauge theories on noncommutative spacetime. To discuss them all is beyond of the scope of this theses, the interested reader may follow the discussion in [Zah06].

One approach to handle noncommutative gauge theories is the use of the Seiberg Witten map, where the noncommutative U(N) theory is mapped to a commutative U(N) theory [SW99]. Roughly speaking it gives an expression of the noncommutative field in terms of the commutative field.

Im Dokument Quantum Field Theory and Composite Fermions in the Fractional Quantum Hall Effect (Seite 115-119)