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arXiv:1111.3050v2 [math-ph] 18 Nov 2011

A numerical approach to harmonic non-commutative spectral

field theory

Bernardino Spisso1and Raimar Wulkenhaar2

1,2Mathematisches Institut der Westf¨ alischen Wilhelms-Universit¨ at

Einsteinstraße 62, D-48149 M¨ unster, Germany

Abstract

We present a first numerical investigation of a non-commutative gauge theory defined via

the spectral action for Moyal space with harmonic propagation. This action is approximated

by finite matrices. Using Monte Carlo simulation we study various quantities such as the

energy density, the specific heat density and some order parameters, varying the matrix size

and the independent parameters of the model. We find a peak structure in the specific heat

which might indicate possible phase transitions. However, there are mathematical arguments

which show that the limit of infinite matrices is very different from the original spectral model.

1Introduction

Quantum field theory on noncommutative spaces [1, 2, 3] is an active subject of research. The

most-studied noncommutative spaces are the Moyal space [4] and fuzzy spaces [5]. Fuzzy spaces

are matrix approximations of manifolds and as such ideal for numerical investigations similar to

non-perturbative quantum field theory on the lattice. In this paper we focus on the Moyal space,

which is a continuous deformation of Rdfor which the usual Fourier techniques of perturbative

quantum field theory are available. It turned out that a renormalisable quantum field theory on

Rdis, in most cases, no longer renormalisable on d-dimensional Moyal space due to a phenomenon

called ultraviolet/infrared mixing [6].In [7] it was discovered that for the ϕ4-model on 4-

dimensional Moyal space the UV/IR-mixing generates an additional marginal coupling which

corresponds to a harmonic oscillator potential for the free scalar field. The resulting action

S[ϕ] =

?

d4x

?1

2ϕ ⋆ (−∆ + Ω2˜ x2+ µ2) ⋆ ϕ +λ

4ϕ ⋆ ϕ ⋆ ϕ ⋆ ϕ

?

(x)(1)

was then shown to be perturbatively renormalisable to all orders in λ. In (1), ˜ x = 2Θ−1·x, where

Θ is the deformation matrix defining the Moyal product. See also [8, 9, 10, 11, 12]. Moreover,

the frequency parameter can be restricted to Ω ∈ [0,1] by Langmann-Szabo duality [13].

1nispisso@tin.it

2raimar@math.uni-muenster.de

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The renormalisability of the action (1) raises the question whether a harmonic term can also

render Yang-Mills theory renormalisable on Moyal space (recall that the usual Yang-Mills action

on Moyal space has the same UV/IR-mixing problem [14]). Yang-Mills theories in noncommut-

ative geometry [15] are naturally obtained from the spectral action principle [16] relative to an

appropriate Dirac operator. In [17] it was shown that Moyal space (with usual Dirac operator)

is a (non-compact) spectral triple; its corresponding spectral action was computed in [18]. To

obtain a gauge theory with harmonic oscillator potential via the spectral action principle, a

differential square root of the harmonic oscillator hamiltonian is necessary as Dirac operator.

In absence of such a Dirac operator, in [19, 20] an effective gauge model was constructed as the

one-loop effective action of complex harmonic noncommutative quantum ϕ4-theory in a classical

external gauge field. As a result, the noncommutative Yang-Mills Lagrangian is extended by

two terms Xµ⋆ Xµand (Xµ⋆ Xµ)2, where Xµ=1

2˜ xµ+ Aµis the ‘covariant coordinate’.

A first outline of a candidate spectral triple for harmonic oscillator Moyal space was given

in [21]. Additionally, in [21] the linear and quadratic terms of the spectral action for a U(1)-

Yang-Mills-Higgs model were computed and then extended by gauge invariance. Thereby the

appearance of Xµ⋆ Xµwas traced back to a deep entanglement of gauge and Higgs fields in a

unified potential (αXµ⋆ Xµ+ β ¯ ϕ ⋆ ϕ − 1)2, with α,β ∈ R+.

It turned out that the candidate spectral triple proposed in [21] was the shaddow of a

new class of non-compact spectral triples with finite volume [22]. The spectral geometry of

Moyal space with harmonic propagation, which falls into this class, was fully worked out in

[23]. There are in fact two (even, real) spectral triples (A,H,D•,Γ,J), with • ∈ {1,2}, for

the d-dimensional Moyal algebra A and differential quare roots D•of the harmonic oscillator

hamiltonian. The spectral triples are of metric dimension d and KO-dimension 2d, have simple

dimension spectrum consisting of the integers ≤ d, and satisfy all regularity and compatibility

requirements of spectral triples. Additionally, the spectral action was rigourously computed in

[23], i.e. with H¨ older type estimates for the remainder of the asymptotic expansion and with

inclusion of the real structure J.

A completely new feature of the spectral action [23, 21] (and also of the effective action

[19, 20]) is that the expansion of Xµ⋆ Xµand its square produces a term which is linear in

the gauge field A. This means that the vacuum, i.e. the solution of the classical field equations,

is no longer taken at Aµ = 0 (or more generally at a flat connection Fµν = 0) but at some

non-constant value for the gauge field. A first discussion of the vacuum structure of this type of

gauge models was given in [24]. It turned out that generically there are infinitely many vacuum

solutions. Some of them were exposed, but it was not possible to give reasonable argument for

the right solution. In particular, it became completely impossible to study the gauge model as

a perturbative quantum field theory.

This is the point where the numerical treatment comes into play. The standard method

of numerical quantum field theory is to approximate the space by discrete points, for example

using a lattice approximation and then calculate the observables over that set of points [25]. For

Moyal space a position space approximation is not suitable due to the oscillator factor of the

Moyal product. Instead, we shall use the matrix Moyal base (which was already used in the first

renormalisation proof [7] of ϕ4-model), restrict it to finite matrices and perform a Monte Carlo

simulation of the resulting action. In this way we will study some statistical quantities such as

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energy density and specific heat, varying the parameters Ω,χ−1

some information on the various contributions of the fields to the action. The simulations are

quite cumbersome due the complexity of the action and the number of independent matrices

to handle. Nevertheless we are able to get an acceptable balance between the computation

precision and the computation time. For the simulations we apply a standard Metropolis-Monte

Carlo algorithm [26] with various estimators for the error and for the autocorrelation time of the

samples. The range of parameters is chosen to avoid problems with the thermalisation process,

thus permitting a relative small number of Monte Carlo steps to compute independent results

from the initial conditions.

χ0,α of the model and gathering

We are eventually interested in the continuum limit which corresponds to matrices of infinite

size. We thus compute our observables such as the energy density for various matrix sizes and

then look for a stabilisation of these observables as the matrix size increases. The specific

heat, which is a measure of the dispersion of the energy, will be used to identify possible phase

transitions in form of peaks of the specific heat at increasing matrix size.

2 Four-dimensional harmonic Yang-Mills model

The harmonic Yang-Mills model is defined as the spectral action resulting from the spectral

triples (A⋆,H,D•,Γ,J), with • ∈ {1,2}, analysed in [23]. The Moyal algebra A⋆is the space of

Schwartz class functions on R4equipped with the product

f ⋆ g(x) =

?

R4×R4

dy dk

(2π)4f(x+1

2Θ · k)g(x+y)ei?k,y?. (2)

The unbounded selfadjoint operators D• on the Hilbert space H are differential square roots

of the harmonic oscillator hamiltonian H = −∂µ∂µ+˜Ω2xµxµof frequency˜Ω, i.e. D2

(−1)•˜ΩΣ, for a certain spin matrix Σ. If L⋆(f) denotes left Moyal multiplication with a function

f ∈ A⋆, then one has

[D1,L⋆(f)] = L⋆(i∂µf) ⊗ Γµ,

•= H −

[D2,L⋆(f)] = L⋆(i∂µf) ⊗ Γµ+4, (3)

where the matrices Γ1,...,Γ8satisfy the anticommutation relations

{Γµ,Γν} = {Γµ+4,Γν+4} = 2(g−1)µν,{Γµ,Γν+4} = 0 ,(4)

relative to an induced metric g = (id −1

real structure satisfy JD•J−1= D•and JL⋆(f)J−1= R⋆(¯f), where R⋆denotes right Moyal

multiplication.

4˜Ω2Θ2)−1. The grading is Γ = Γ1···Γ8, and the

In order to implement the Higgs mechanism ` a la Connes-Lott [27] one considers the product

of the spectral triple (A⋆,H,D1,Γ,J) with the finite Higgs spectral triple (C⊕C,C2,Mσ1,Jf),

where σ1is a Pauli matrix, Jf any matricial real structure and M > 0. Then, a self-adjoint

fluctuation A =

?ai[D,bi] of the total Dirac operator D = (D1⊗ 1 + Γ ⊗ Mσ1) to give

DA= D + A + JAJ−1, for J = J ⊗ Jf, is of the form

A =

?

ΓµL⋆(Aµ)

ΓL⋆(¯φ)

ΓL⋆(φ)

ΓµL⋆(Bµ)

?

,(5)

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for two real one-forms Aµ,Bµ ∈ A⋆ and a complex scalar φ ∈ A⋆. Using D2 instead of D1

amounts to replace Γµby Γµ+4.

The spectral action principle [16] asserts that the bosonic action of a field theory with

fermionic Dirac operator DAhas the form

S(DA) = Tr(χ(D2

where χ is a smooth approximation of the characteristic function on [0,Λ2], for some scale

parameter Λ. For the fluctuation (5), the part of the spectral action which is relevant and

marginal for Λ → ∞ has been explicitly computed in [23], for general effective metric g. This

computation involved Laplace transformation, Duhamel expansion with H¨ older-type estimates

for the remainder and explicit use of the Mehler kernel for the harmonic oscillator hamiltonian.

For a special choice of the noncommutativity matrx Θ2= −θ2id, the result of [23] takes in

terms of Ω :=θ˜Ω

χ0:= χ(0) the form

SΛ(DA) =θ4Λ8

8Ω4

16Ω4

?52

+

π2(1 + Ω2)2

4Ω2

1+Ω2˜ XAµ⋆˜ Xµ

4Ω2

1+Ω2˜ XBµ⋆˜ Xµ

?(1 + Ω2)2

+ O(Λ−1) .

Here, Dµφ = ∂µφ − iAµ⋆ φ + iφ ⋆ Bµ− iM(Aµ− Bµ) is the covariant derivative of the scalar

field, FA

the field strength of B. Moreover,˜ XAµ:=˜ X0µ+ Aµand˜ XBµ:=˜ X0µ+ Bµare the covariant

derivatives of A and B, respectively, where˜ X0µ:=

of the spectral action (7) is that the Higgs field φ and the gauge fields A,B appear together

in a unified potential. In this way, also the gauge field shows a non-trivial vacuum structure.

Besides, the action is invariant under U(A⋆) × U(A⋆) transformations:

(φ + M) ?→ uA⋆ (φ + M) ⋆ uB,

A)) , (6)

2, the moments Λ2nχ−n:=?∞

8Ω4χ−4−M2θ4Λ6

45+M8θ4

?

φ⋆¯φ + M(φ+¯φ) +

0ds sn−1χ(s) of the “characteristic function” and

χ−3+

?M4θ4Λ4

χ0

+8θ2Λ4

12Ω2

?

χ−2−

?M6θ4Λ2

48Ω4

+2M2θ2Λ2

3Ω2

?

χ−1

+

192Ω4+M4θ2

?

3Ω2

?

χ0

d4x2(1 + Ω2)Dµφ ⋆ Dµφ

+

?

?¯φ⋆φ + M(φ+¯φ) +

A+ M2−χ−1

B+ M2−χ−1

χ0

Λ2?2

Λ2?2

−

?4Ω2

?4Ω2

1+Ω2˜ X0µ⋆˜ Xµ

0+ M2−χ−1

0+ M2−χ−1

χ0

Λ2?2

Λ2?2

+

χ0

−

1+Ω2˜ X0µ⋆˜ Xµ

χ0

+

2

−(1 − Ω2)4

6(1 + Ω2)2

??FA

µν⋆ FAµν+ FB

µν⋆ FBµν??

(7)

µν:= ∂µAν− ∂νAµ− i(Aµ⋆ Aν− Aν⋆ Aµ) the field strength of A and similarly FB

µν

˜ xµ

2= (Θ−1)µνxν. The remarkable outcome

˜ XA

µ?→ uA⋆˜ Xµ

A⋆ uA,

˜ Xµ

B?→ uB⋆˜ Xµ

B⋆ uB.(8)

3Discretisation by Moyal base

The 2-dimensional Moyal algebra with deformation parameter θ > 0 has a natural basis of

eigenfunctions fmn of the harmonic oscillator, where m,n ∈ N.

coordinates by

These are given in radial

fmn(ρcosϕ,ρsinϕ) = 2(−1)m

?

m!

n!eiϕ(n−m)

??

2

θρ

?n−m

e−ρ2

θ Ln−m

m

?2

θρ2

?

(9)

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and satisfy

(fmn⋆ fkl)(x) = δnkfml(x)

?

(10)

d2xfmn(x) = 2πθδmn, (11)

see [4, 17] for details. The expansion of Schwartz functions on R4in the Moyal base,

A⋆∋ a = a(x0,...,x3) =

?

m1,m2,n1,n2∈N

am1n1

m2n2fm1n1(x0,x1)fm2n2(x2,x3) , (12)

then provides an isomorphism of Fr´ echet spaces between A⋆and the space of rapidly decreasing

double sequences (amn)m,n∈N2 equipped with the family of seminorms

pk((amn)m,n∈N2) :=

∞

?

m,n∈N2

?

(2|m| + 1)2k(2|n| + 1)2k|amn|2?1

2,|m| := m1+ m2.(13)

According to (10), Moyal product and integral reduce in the (fmn)-basis to product and trace of

infinite N2-labelled matrices, with convergent index sums due to (13). By duality, the covariant

derivatives XA

µcan also be expanded in the (fmn)-basis, but the expansion coefficients

XA

µm1n1

µm1n1

m2n2

To any a ∈ A⋆we can associate a sequence (aN)N∈Nof cut-off matrices

?

0else .

µand XB

diverge for mi,ni→ ∞.

m2n2, XB

aN

m1n1

m2n2

=

am1n1

m2n2

if max(m1,m2,n1,n2) ≤ N ,

Then, (aN) is a Cauchy sequence in any of the semi-norms pkand converges to a in the Fr´ echet

topology of A⋆.

In quantum field theory we are confronted with the converse problem. To deal with diver-

gences, a regularisation has to be introduced which restricts the system to a finite number of

degrees of freedom. After re-normalisation from bare to physical quantities one has to show that

the limit to an infinite number of degrees of freedom is well-defined. In our case, the natural

regularisation is to restrict the matrix indices to mi≤ N, which corresponds to a cut-off in the

energy. Even if we could solve the renormalisation problem, the removal of the cut-off, i.e. the

limit N → ∞ to infinite matrices, will fail: A sequence of (N × N)-matrix algebras does not

converge in the Fr´ echet topology.

The question of convergence of matrix algebras has been studied in a sequence of articles

[28, 29, 30] by Marc Rieffel, David Kerr and Hanfeng Li [31, 32, 33]. Rieffel defined in [28]

the quantum Gromov-Hausdorff distance for compact quantum metric spaces, which are order-

unit spaces equipped with a generalised Lipschitz seminorm, the Lip-norm. In [29] he proved

that the sequence of matrix algebras which defines the fuzzy sphere converges in the quantum

Gromov-Hausdorff distance to the continuous functions on the sphere. While it seems possible

to give along these lines a reasonable notion of convergence of (N × N)-matrix algebras to a

limiting C∗-algebra (which describes continuity), we see at the moment no possibility to specify

convergence to the Fr´ echet algebra A⋆which describes smoothness.

This lack of convergence in the Fr´ echet topology implies that the results we obtain for finite

approximations of the action (7) are probably of little relevance for the smooth model. Thus,

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