matrix transforms in qubit field theory

M. D. Sheppeard Department of Physics and Astronomy University of Canterbury, Christchurch, New Zealand (Dated: November 2, 2007) Circulant mass matrices for triples of charged and neutral have been studied in the context of qubit quantum field theory. This note describes the discrete Fourier transform behind such matrices, and discusses a category theoretic interpretation of these operators.

PACS numbers: 03.67.-a, 03.67.Lx, 04.60.-m

INTRODUCTION for real η, r and θ. In terms of these parameters, the eigenvalues are given by Using a measurement algebra approach to QFT, Bran- 2πk nen [1] recently recovered the Koide [2] formula λk = η(1 + 2rcos(θ + )) 3 √ √ √ 2 3 ( me + mµ + mτ ) = (me + mµ + mτ ) (1) r2 1 2 The Koide formula (1) follows when = 2 and this choice may be applied also to the matrix. for charged in the form of a 3 × 3 circu- In the n × n case, the discrete Fourier transform [3][4] lant complex matrix, whose eigenvalues squared give the interchanges the set of eigenvalues λk (assumed distinct) lepton masses to experimental precision. This analysis and matrix entries A1,A2,A3, ··· ,An via was extended to a set of three , and the mass  ratio predictions agree with preliminary neutrino oscilla- 2πijk λk = e n Aj (4) tion data. j Here it is observed that the discrete Fourier transform  1 − 2πijk [3] provides a further interpretation of the mass matrices, A e n λ j = n k both as a duality between operators and eigenvalues and k also as a link to the theory of quantum computation [4]. It is expected that other triples of Standard Model Viewing the eigenvalues as a diagonal matrix, the trans- form interchanges the bases of projection operators and particles, namely baryons and mesons, will also be asso- cyclic permutations. For real eigenvalues (m1,m2,m3) ciated with 3 × 3 matrix operators of the same kind in 2 2πi with mi = λi in the above, and letting ω = e 3 ,the accord with their structure [1] and the association transform takes the diagonal matrix to the circulant

of spatial directions to the number of particle genera- ½ ¼ 2 2 tions, given by the three primitive idempotents of the m1 + m2 + m3 m1ω + m2ω + m3 m1ω + m2ω + m3

2 2  measurement algebra.  m1ω + m2ω + m3 m1 + m2 + m3 m1ω + m2ω + m3 2 2 m1ω + m2ω + m3 m1ω + m2ω + m3 m1 + m2 + m3

FOURIER TRANSFORMS AND MASS which must be the square of (3) since the square of a MATRICES circulant is a circulant. Thus a choice of scale is specified 1 by η = 3 (m1 + m2 + m3). × A circulant matrix is built from its first row by adding A3 3 matrix is viewed as a function on the discrete Z × Z cyclic permutations. In particular, a 3×3 circulant takes torus 3 3, which has a quantum description in terms of the convolution product for matrices [3]. Letting Dij = the form i   δijω this product satisfies the Weyl rule ABC   CAB (2) D ∗ (312) = ω(312) ∗ D BCA where the phase 2π is proportional to −1. This asso- where A, B and C will be complex numbers. Note that 3 ciates Planck’s constant with a heirarchy N determined any such circulant is a combination of the three permu- by the size of the matrix, but the continuum limit is ob- tations (123), (231) and (312). For real eigenvalues λk tained via →∞rather than → 0. it is essential that A be real and C = B. Thus a mass If masses are to be thought of as quantum numbers, matrix [1] takes the form   then why are their values so awkward in comparison to, 1 reiθ re−iθ say, spin? For 2 × 2 circulants with entries A and B, C = η  re−iθ 1 reiθ  (3) the eigenvectors are (1, 1) and (1, −1) with eigenvalues reiθ re−iθ 1 (A + B) and (A − B) respectively. For example, for the Pauli swap matrix σx, with A = 0, the spin eigenvalues ables. However, it is seen that phase space variables sat- are ±1. Complexity in the eigenvalue set only arises in isfy the Weyl algebra of the quantum plane. Is there a dimension three or higher. noncommutative transform that extends this analysis to λk Degenerate eigenvalues η ∈{1−r, 1−r, 1+2r} occur nonclassical underlying spaces? This is relevant to the when θ = 0 and all matrix entries are real. Although question of extending the perturbative rest mass com- this pattern does not describe the leptons, we observe putations [1] to a nonperturbative regime dealing with that it is the typical composition of masses for baryon physical scale. constituents. Since such mass operators arise in a preon Kapranov [5] has recently considered path spaces ap- model that unifies particle structure, it is expected that proximated by cubical paths, each of which is represented all standard model bound states and resonances may be by a noncommutative monomial in the spatial directions. arranged into mass triples. In dimension d>1 a noncommutative Fourier transform In quantum computation [4] a Fourier transform is also relates measures on the space of paths to functions of defined in this way, acting on a set of n basis states. An N the noncommuting variables. The basic idea is that a qubit computer has n =2N basis states. The transform path integral is just a map from a noncommutative ring is unitary because it may be built from unitary gates, 1 to a suitable commutative subring. In this way, particle namely the Hadamard gate H = √ (σx + σz) and the 2 masses [1] could arise as path integral invariants. series   Taking T-duality seriously, one also expects to deal 10 Bk = 2πi with nonassociativity. From a category theoretic point 0 e 2k of view, both noncommutative and nonassociative struc- tures can be dealt with in a unified framework. The N By analogy, a mass computation with 3 basis states cohomological element of interest here is the parity cube uses ternary digits, so the gates Bk would be replaced by axiom, which describes the now familiar pentagon law gates on five of its faces. In a sufficiently lax algebraic setting,   such as for tetracategories, the sixth face may break this 10 0  2πi  law, providing the deformation parameter that turns a Tk =  0 e 3k 0  (5) pentagon into a hexagon representing the permutation 4πi 00e 3k group S3 [6]. which are also unitary. It is not clear, however, what op- The generation count by primitive idempotents [1] is erator should replace the Hadamard gate H, since it has confirmed by the string theoretic index theorem argu- ment applied to the Riemann moduli space of the six no 3 × 3 analogue. Noting that σz carries the spin eigen- values, one possibility is a combination of mass diago- punctured sphere, which has an orbifold Euler character- nals and the generator (312), but such an operator is not istic [7] of -6. The six punctures are associated to the unitary. Such non-unitary processes would have major six faces of a cube via a dual vertex, which is thickened implications for the black hole paradox, which may only to a sphere. Note that cohomological integrals for such conserve qubit (flat space) information, but not gravita- moduli spaces commonly appear in QFT computations tional information. On the other hand, the meaning of as multiple zeta values and polylogarithms. time itself is altered in this approach, which does not as- For helpful discussions I thank Carl Brannen, Michael sume a globally defined time for a nonsensical universal Rios, Matti Pitkanen, Tony Smith and Louise Riofrio. observer. Note the similarity between the Tk and powers of the diagonal D that appears in the Weyl relations. Given the direct application of the qubit Fourier transform to num- ber factoring, this associates ternary factorisation with [1] C. A. Brannen, http://brannenworks.com/dmaa.pdf. 34 the quantum torus. [2] Y. Koide, Lett. Nuovo Cim. , 201 (1982). [3] R. Aldrovandi, Special matrices of mathematical physics (World Scientific, 2001). Quantum computation DISCUSSION [4] M. A. Nielsen and I. L. Chuang, and quantum information (Cambridge, 2000). [5] M. Kapranov, math.QA/0612411. The mass matrices arise from a one dimensional dis- [6] M. Batanin, math.CT/0301221. crete transform, which itself involves commutative vari- [7] M. Mulase and M. Penkava, math-ph/9811024.