Generalized Wigner distributions and scalar field on a fuzzy sphere Juraj Tekel The Graduate Center and The City College The City University of New York work with V.P. Nair and A. Polychronakos Models of Modern Physics, Svit 7. 9. 2011
Formulation of the problem Lets have an ensemble of random matrices with the following probability measure ) ) 1 1 1 (M 2 ( Z e 2 ν 2 Tr M[L α,[l α,m]] 1 2 µ2 Tr M - hermitian N N random matrix L α and α = 1, 2, 3 - SU(2) generators in corresponding representation ν, µ - parameters Z - normalization In other words f (M) = dm 1 Z e 1 2 ( ) ) 1 ν 2 Tr M[L α,[l α,m]] 1 2 µ2 Tr (M 2 f (M)
What are the expected values of some quantities of interest, Tr (M), Tr ( M k), Tr ( [L α, [L α, M]] k),...? What is the distribution of the eigenvalues of M and functions of M?
The physical motivation will be discussed in more detail in a while. But since the L α s are the angular momentum operators and SU(2) is the symmetry group of the sphere, we see that in ) ) 1 1 1 (M 2 ( Z e 2 ν 2 Tr M[L α,[l α,m]] 1 2 µ2 Tr the first term is like a kinetic term and the second term is like a mass term for a field M on something like a sphere. More on that later.
The Wigner distribution Measure without the first term, namely 1 Z e 1 2 µ2 Tr(M 2 ) gives a standard and well known result Differences full SU(N) symmetry observables only powers of M the angular part is not important
Solution diagonalize M as UΛU 1, with Λ = diag(λ 1,..., λ N ) diagonal matrix of the eigenvalues the integration measure becomes dm = du 2 (λ) is Vandermonde determinant N i=1 λ i (λ) = i<j (λ i λ j ) effective action (or measure) S eff = 1 2 µ2 λ 2 i 2 i<j log λ i λ j
gas of N-particles in an effective potential we define the eigenvalue density as ρ(λ) = i δ(λ λ i ) this becomes continuous as N writing down equations of motion we can find an equation for ρ we use the fact that in the large N limit ( Tr M k) = dλλ k ρ(k)
solving the equation we find ( ) ρ(λ) = 2µ2 N N N π µ 2 λ2 λ µ, µ rescaling the original matrix M M/ N this becomes finite in the large-n limit this is the Wigner semicircle law
Back to our problem ( ) ) 1 1 1 Z e 2 ν 2 Tr M[L α,[l α,m]] 1 2 µ2 Tr (M 2 We use a different approach We write down recursion rules for Tr ( M a B b) in terms of lower order traces and two point functions of the theory Rewrite these in terms of the generating function φ(t, s) = ( Tr M a B b) t a s b a,b and solve for φ We get a joint probability distribution for eigenvalues of M and B As before, the analysis is in the large N limit
This approach uses only the two point functions of the theory MM, MB, BB and therefor is applicable to a more general case of the action We expand the matrix M = ca l T A l in terms of basis T l A, l = 0, 1,..., N 1, A = 1, 2,..., 2l + 1 normalized as ( Tr T l A T l B ) = δ ll δ AB TA l T A l ij = 2l + 1 N δ ij
The action becomes S = A,l [ 1 1 2 ν 2 l(l + 1) + 1 ] 2 µ2 c l A and therefor MM ij = 1 N (2l + 1)G(l)δij = f δ ij MB ij = 1 N (2l + 1)l(l + 1)G(l)δij = hδ ij BB ij = 1 N (2l + 1)l 2 (l + 1) 2 G(l)δ ij = gδ ij where G(l) is the l-dependent part of the propagator. In our case it is ν 2 G(l) = l(l + 1) + µ 2 ν 2
To obtain finite expression we redefine M = 1 f M, B = 1 g B then MM ij = δ ij, MB ij = γδ ij, BB ij = δ ij with only one relevant parameter γ = h fg and W a,b = 1 N Tr [ ( M f ) a ( ) ] B b g, φ(t, s) = W a,b t a s b
The recursion rules for Wa,b
We recover the same equations also as Schwinger-Dyson equations of the model, i.e. equations of motion for the corresponding Green s functions. 4W a,b = 4W a,b = a 2 b 1 W α,0 W a 2 α,b + γ W a 1,b α 1 W 0,α α=0 α=0 b 2 a 1 W 0,α W a,b 2 α + γ W a α 1,b 1 W α,0 α=0 α=0
Solution φ(t, s) = 4 (1 + 1 t 2 ) (1 + 1 t 2 ) γts and defining the two dimensional distribution W a,b = dxdyρ(x, y)x a y b we obtain 1 γ 2 ρ(x, y) = ρ(x)ρ(y) (1 γ 2 ) 2 4γ(1 + γ 2 )xy + 4γ 2 (x 2 + y 2 ) positive for γ 2 < 1 limits γ 1 ρ(x, y) = ρ(x)δ(x y), γ 1 ρ(x, y) = ρ(x)δ(x+y) for the case γ = 0 two independent Wigner distributions
For the density of the eigenvalues of M, we recover the Wigner distribution with radius ( ν 2 log 1 + 1 ) µ 2 ν 2 This becomes the original radius in the limit ν O Connor and Saemann computed the same quantity with the kinetic term as a perturbation to the kinetic term. They found a polynomial correction to the Wigner distribution. Probably indicates that the zero ν limit can not be continuously connected to the full model and a possible phase transition.
Scaling of G(l) For large l let G(l) l α. Then f = 1 (2l + 1)G(l)δij = 2 l α+1 N N h = 1 2 (2l + 1)l(l + 1)G(l) = l α+3 N N g = 1 (2l + 1)l 2 (l + 1) 2 G(l) = 2 l α+5 N N then for γ = h/ fg we have possible cases α > 2 then γ = const 4 < α < 2 then γ << 1 6 < α < 4 then γ << 1 α < 6 then γ = const
Outlook Calculation of distributions for different observables ( Tr M a B b M c B d) ( ), Tr M a [L α, M]M b [L α, M],... More general actions Different spaces, NC torus, NC projective spaces, etc.
Why is this interesting for a physicist? As mentioned, something like a field theory on something like a sphere.
Fuzzy sphere the regular sphere is just a set of points in 3D-space x i x i = R 2, x i x j x j x i = 0 coordinates constrained in such way generate the algebra of functions on this regular sphere we define the fuzzy sphere to be a space, for which the algebra of functions is generated by generators constrained by x i x i = ρ 2, x i x j x j x i = iε ijk x k coordinates do not longer commute non-commutative for each j we have a space = fuzzy sphere, in the limit j we recover the regular sphere realized by spin-j representation of SU(2), then the radius becomes j(j + 1)
so coordinates on fuzzy sphere can be viewed as SU(2) matrices of dimension N = 2j + 1 the fields become general N N matrices, derivatives (small rotations) become L-commutators (space-time) integration becomes trace euclidean field theory is defined by the action, e.g. for a free field S = 1 2 Tr( [L α, [L α, M]] ) + 1 2 µ2 Tr ( M 2) and the functional correlation functions F[M] = dm e S F[M] dm e S
Therefor our results can be directly used for computation of correlation functions in the theory of a scalar field on the fuzzy sphere, or different fuzzy spaces.
Why is this (NC field theory) interesting for a physicist? finite number of modes space-time becomes finitely structured at short distance without breaking the symmetry regularization effective description of quantum Hall systems role in quantum gravity string theories
A (completely) different question Given a vacuum state 0 and an observable A, what is the probability of measuring the value α of this observable in this vacuum? this probability density reflects the vacuum fluctuations of the observable A
A good guess is P(α)dα = dα 0 δ(α A) 0 Since now P(α)dα = 1 αp(α)dα = Ā α 2 P(α)dα = Ā2 etc.
We expand the delta function P(A) = 0 δ(α A) 0 = 1 dλ 0 e iαλ iλa 0 = 2π = 1 dλe iαλ ( iλ) n A n 2π n! n
We try to look for energy fluctuations. Large enough fluctuations should produce a black hole. If the rate of this production was observable, this would suggest an inconsistency of the combination of the general relativity and quantum field theory. A possible new insight into the cosmological constant problem.
An aside - The cosmological constant problem Ground state of quantum harmonic oscillator has energy E = 1 2 ω Quantum field is equivalent to collection of harmonic oscillators with ω k = k 2 + m 2 So the ground state has energy density ρ = d 3 k (2π) 3 1 2 ω k
After regularization, this is proportional to with ε the cutoff distance. ρ ε 4 And with some nice story this is usually discarded... however...
The stories do not work if we work in curved space-time. Here the action is S = d 4 x gl(φ, φ) and constant shift in energy is no longer a constant term in the action. It effects equations of motion for the metric itself.
For the Einstein-Hilbert action S = d 4 x [ ] 1 g R + L(φ, φ) 16πG we get equations of motion for the space-time metric R µν 1 2 g µνr + 8πG ρ g µν = 8πGT µν So the vacuum energy is equivalent to a cosmological constant.
We now believe that the universe has positive cosmological constant, because the expansion of universe is accelerating (distant supernovae) repulsive gravity of Λ the universe is flat (CMB anisotropies) but baryons and dark matter make only 30% of critical density
To predict the value, we take the cutoff to be Plank length and get the corresponding energy density ρ 10 71 GeV 4 Value of the cosmological constant energy density consistent with the measurements is ρ 10 47 GeV 4
The difference is 118 ORDERS OF MAGNITUDE!!!
If the rate of the creation of the black holes in vacuum would be an observable number, the vacuum energy could still be the source of the cosmological constant, invalidating the previous huge discrepancy.
back to the probabilities The relevant observable is an energy E in a given volume V A = d 3 xt 00 (x) and we consider the case of a free massive scalar field S = 1 2 ( φ)2 1 2 m2 φ 2 and A = V V d 3 x [( 0 φ) 2 12 ] ( φ)2 Integrals are going to be divergent and we need to regularize
We regularize by considering the space-time to be non-commutative. Our previous results are going to be useful here, however there are still some issues to solve, both technical and conceptual. Ďakujem za pozornost