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Journal articles on the topic 'Yang-Mills matrix model'

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Published: 6 September 2023

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1

Aoki, Hajime, Nobuyuki Ishibashi, Satoshi Iso, Hikaru Kawai, Yoshihisa Kitazawa, and Tsukasa Tada. "Non-commutative Yang–Mills in IIB matrix model." Nuclear Physics B 565, no. 1-2 (January 2000): 176–92. http://dx.doi.org/10.1016/s0550-3213(99)00633-1.

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2

Pandey, Mahul, and Sachindeo Vaidya. "Yang–Mills matrix mechanics and quantum phases." International Journal of Geometric Methods in Modern Physics 14, no. 08 (May 11, 2017): 1740009. http://dx.doi.org/10.1142/s0219887817400096.

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The [Formula: see text] Yang–Mills matrix model coupled to fundamental fermions is studied in the adiabatic limit, and quantum critical behavior is seen at special corners of the gauge field configuration space. The quantum scalar potential for the gauge field induced by the fermions diverges at the corners, and is intimately related to points of enhanced degeneracy of the fermionic Hamiltonian. This in turn leads to superselection sectors in the Hilbert space of the gauge field, the ground states in different sectors being orthogonal to each other. The [Formula: see text] Yang–Mills matrix model coupled to two Weyl fermions has three quantum phases. When coupled to a massless Dirac fermion, the number of quantum phases is four. One of these phases is the color-spin locked phase. This paper is an extended version of the lectures given by the second author (SV) at the International Workshop on Quantum Physics: Foundations and Applications, Bangalore, in February 2016, and is based on [1].

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3

Siegel, W. "Super Yang-Mills theory as a random matrix model." Physical Review D 52, no. 2 (July 15, 1995): 1035–41. http://dx.doi.org/10.1103/physrevd.52.1035.

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4

Kimura, Yusuke, and Yoshihisa Kitazawa. "Supercurrent interactions in noncommutative Yang–Mills and IIB matrix model." Nuclear Physics B 598, no. 1-2 (March 2001): 73–86. http://dx.doi.org/10.1016/s0550-3213(00)00785-9.

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5

Xu, Feng. "A Random Matrix Model From Two Dimensional Yang-Mills Theory." Communications in Mathematical Physics 190, no. 2 (December 1, 1997): 287–307. http://dx.doi.org/10.1007/s002200050242.

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6

Acharyya, Nirmalendu, A. P. Balachandran, Mahul Pandey, Sambuddha Sanyal, and Sachindeo Vaidya. "Glueball spectra from a matrix model of pure Yang–Mills theory." International Journal of Modern Physics A 33, no. 13 (May 9, 2018): 1850073. http://dx.doi.org/10.1142/s0217751x18500732.

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We present variational estimates for the low-lying energies of a simple matrix model that approximates SU(3) Yang–Mills theory on a three-sphere of radius R. By fixing the ground state energy, we obtain the (integrated) renormalization group (RG) equation for the Yang–Mills coupling g as a function of R. This RG equation allows to estimate the mass of other glueball states, which we find to be in excellent agreement with lattice simulations.

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7

CHAN, HONG-MO. "YANG–MILLS DUALITY AS THE ORIGIN OF FERMION GENERATIONS." Modern Physics Letters A 18, no. 08 (March 14, 2003): 537–43. http://dx.doi.org/10.1142/s0217732303009629.

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A non-Abelian extension of electric-magnetic duality implies that dual to confined colour SU(3), there also ought to be a broken threefold symmetry which can play the role of fermion generations. A model constructed on these premises not only gives a raison d'être for 3 and only 3 generations as observed but also offers a natural explanation for the distinctive fermion mass and mixing patterns seen in experiment. A calculation to one-loop order in this model with only 3 fitted parameters already gives correct values, all within present experimental errors, for the following quantities: the mass ratios mc/mt, ms/mb, mμ/mτ, all 9 matrix elements of the CKM mixing matrix |Vrs| for quarks, plus the lepton MNS mixing matrix elements |Uμ3| and |Ue3| studied in neutrino oscillation experiments with respectively atmospheric and reactor neutrinos.

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8

Pandey, Mahul, and Sachindeo Vaidya. "Quantum phases of Yang-Mills matrix model coupled to fundamental fermions." Journal of Mathematical Physics 58, no. 2 (February 2017): 022103. http://dx.doi.org/10.1063/1.4976503.

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9

Kawamoto, Shoichi, and Dan Tomino. "A renormalization group approach to a Yang–Mills two matrix model." Nuclear Physics B 877, no. 3 (December 2013): 825–51. http://dx.doi.org/10.1016/j.nuclphysb.2013.10.021.

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10

PANZERI, STEFANO. "THE c=1 MATRIX MODEL FORMULATION OF TWO-DIMENSIONAL YANG-MILLS THEORIES." Modern Physics Letters A 08, no. 33 (October 30, 1993): 3201–14. http://dx.doi.org/10.1142/s0217732393002130.

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We find the exact matrix model description of two-dimensional Yang-Mills theories on a cylinder or on a torus and with an arbitrary semisimple compact gauge group. This matrix model is the singlet sector of a c=1 matrix model where the matrix field is in the fundamental representation of the gauge group. We also prove that the basic constituents of the theory are Sutherland fermions in the zero coupling limit, and this leads to an interesting connection between two-dimensional gauge theories and one-dimensional integrable systems. In particular we derive for all the classical groups the exact grand canonical partition function of the free fermion system corresponding to a two-dimensional gauge theory on a torus.

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11

Szabo, Richard J., and Miguel Tierz. "Chern–Simons matrix models, two-dimensional Yang–Mills theory and the Sutherland model." Journal of Physics A: Mathematical and Theoretical 43, no. 26 (June 4, 2010): 265401. http://dx.doi.org/10.1088/1751-8113/43/26/265401.

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12

ARNONE, S., G. DI SEGNI, M. SICCARDI, and K. YOSHIDA. "${\mathcal N}=1^*$ MODEL SUPERPOTENTIAL REVISITED (IR BEHAVIOR OF ${\mathcal N}=4$ LIMIT)." International Journal of Modern Physics A 22, no. 28 (November 10, 2007): 5089–115. http://dx.doi.org/10.1142/s0217751x07037998.

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The one-loop contribution to the superpotential, in particular the Veneziano–Yankielowicz potential in [Formula: see text] supersymmetric Yang–Mills model is discussed from an elementary field theory method and the matrix model point of view. Both approaches are based on the renormalization group variation of the superconformal [Formula: see text] supersymmetric Yang–Mills model.

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13

Kogan, I. I., R. J. Szabo, and G. W. Semenoff. "D-Brane Configurations and Nicolai Map in Supersymmetric Yang–Mills Theory." Modern Physics Letters A 12, no. 03 (January 30, 1997): 183–93. http://dx.doi.org/10.1142/s0217732397000182.

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We discuss some properties of a supersymmetric matrix model that is the dimensional reduction of supersymmetric Yang–Mills theory in 10 dimensions and which has been recently argued to represent the short-distance structure of M-theory in the infinite momentum frame. We describe a reduced version of the matrix quantum mechanics and derive the Nicolai map of the simplified supersymmetric matrix model. We use this to argue that there are no phase transitions in the large-N limit, and hence that S-duality is preserved in the full 11-dimensional theory.

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14

POPOV, ALEXANDER D. "BOUNCES/DYONS IN THE PLANE WAVE MATRIX MODEL AND SU(N) YANG–MILLS THEORY." Modern Physics Letters A 24, no. 05 (February 20, 2009): 349–59. http://dx.doi.org/10.1142/s0217732309030163.

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We consider SU (N) Yang–Mills theory on the space ℝ × S3 with Minkowski signature (-+++). The condition of SO(4)-invariance imposed on gauge fields yields a bosonic matrix model which is a consistent truncation of the plane wave matrix model. For matrices parametrized by a scalar ϕ, the Yang–Mills equations are reduced to the equation of a particle moving in the double-well potential. The classical solution is a bounce, i.e. a particle which begins at the saddle point ϕ = 0 of the potential, bounces off the potential wall and returns to ϕ = 0. The gauge field tensor components parametrized by ϕ are smooth and for finite time, both electric and magnetic fields are nonvanishing. The energy density of this non-Abelian dyon configuration does not depend on coordinates of ℝ × S3 and the total energy is proportional to the inverse radius of S3. We also describe similar bounce dyon solutions in SU (N) Yang–Mills theory on the space ℝ × S2 with signature (-++). Their energy is proportional to the square of the inverse radius of S2. From the viewpoint of Yang–Mills theory on ℝ1,1 × S2 these solutions describe non-Abelian (dyonic) flux tubes extended along the x3-axis.

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15

CHEN, B., H. ITOYAMA, and H. KIHARA. "NON-ABELIAN BERRY PHASE, YANG–MILLS INSTANTON AND USp(2k) MATRIX MODEL." Modern Physics Letters A 14, no. 13 (April 30, 1999): 869–77. http://dx.doi.org/10.1142/s0217732399000924.

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The non-Abelian Berry phase is computed in the T dualized quantum mechanics obtained from the USp (2k) matrix model. Integrating the fermions, we find that each of the space–time points [Formula: see text] is equipped with a pair of su(2) Lie algebra valued pointlike singularities located at a distance m(f) from the orientifold surface. On a four-dimensional paraboloid embedded in the five-dimensional Euclidean space, these singularities are recognized as the BPST instantons.

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16

Balachandran, A. P., Sachindeo Vaidya, and Amilcar R. de Queiroz. "A matrix model for QCD." Modern Physics Letters A 30, no. 16 (May 12, 2015): 1550080. http://dx.doi.org/10.1142/s0217732315500807.

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Gribov's observation that global gauge fixing is impossible has led to suggestions that there may be a deep connection between gauge fixing and confinement. We find an unexpected relation between the topological nontriviality of the gauge bundle and colored states in SU (N) Yang–Mills theory, and show that such states are necessarily impure. We approximate QCD by a rectangular matrix model that captures the essential topological features of the gauge bundle, and demonstrate the impure nature of colored states explicitly. Our matrix model also allows the inclusion of the QCD θ-term, as well as to perform explicit computations of low-lying glueball masses. This mass spectrum is gapped. Since an impure state cannot evolve to a pure one by a unitary transformation, our result shows that the solution to the confinement problem in pure QCD is fundamentally quantum information-theoretic.

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17

HSU, JONG-PING. "A UNIFIED GRAVITY-ELECTROWEAK MODEL BASED ON A GENERALIZED YANG–MILLS FRAMEWORK." Modern Physics Letters A 26, no. 23 (July 30, 2011): 1707–18. http://dx.doi.org/10.1142/s021773231103619x.

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Gravitational and electroweak interactions can be unified in analogy with the unification in the Weinberg–Salam theory. The Yang–Mills framework is generalized to include spacetime translational group T(4), whose generators Tμ ( = ∂/∂xμ) do not have constant matrix representations. By gauging T(4) × SU (2) × U (1) in flat spacetime, we have a new tensor field ϕμν which universally couples to all particles and anti-particles with the same constant g, which has the dimension of length. In this unified model, the T(4) gauge symmetry dictates that all wave equations of fermions, massive bosons and the photon in flat spacetime reduce to a Hamilton–Jacobi equation with the same "effective Riemann metric tensor" in the geometric-optics limit. Consequently, the results are consistent with experiments. We demonstrated that the T(4) gravitational gauge field can be quantized in inertial frames.

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18

SATHIAPALAN, B. "THE HAGEDORN TRANSITION AND THE MATRIX MODEL FOR STRINGS." Modern Physics Letters A 13, no. 26 (August 30, 1998): 2085–94. http://dx.doi.org/10.1142/s0217732398002205.

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We use the matrix formalism to investigate what happens to strings above the Hagedorn temperature. We show that is not a limiting temperature but a temperature at which the continuum string picture breaks down. We study a collection of N D-0-branes arranged to form a string having N units of light-cone momentum. We find that at high temperatures the favored phase is one where the string worldsheet has disappeared and the low-energy degrees of freedom consists of N2 massless particles ("gluons"). The nature of the transition is very similar to the deconfinement transition in large-N Yang–Mills theories.

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19

KAWAI, HIKARU, and MATSUO SATO. "PERTURBATIVE VACUA FROM IIB MATRIX MODEL." International Journal of Modern Physics A 23, no. 14n15 (June 20, 2008): 2279–80. http://dx.doi.org/10.1142/s0217751x08041086.

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It has not been clarified whether a matrix model can describe various vacua of string theory. In this talk, we show that the IIB matrix model includes type IIA string theory1. In the naive large N limit of the IIB matrix model, configurations consisting of simultaneously diagonalizable matrices form a moduli space, although the unique vacuum would be determined by complicated dynamics. This moduli space should correspond to a part of perturbatively stable vacua of string theory. Actually, one point on the moduli space represents type IIA string theory. Instead of integrating over the moduli space in the path-integral, we can consider each of the simultaneously diagonalizable configurations as a background and set the fluctuations of the diagonal elements to zero. Such procedure is known as quenching in the context of the large N reduced models. By quenching the diagonal elements of the matrices to an appropriate configuration, we show that the quenched IIB matrix model is equivalent to the two-dimensional large N [Formula: see text] super Yang-Mills theory on a cylinder. This theory is nothing but matrix string theory and is known to be equivalent to type IIA string theory. As a result, we find the manner to take the large N limit in the IIB matrix model.

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20

FERRARI, FRANK. "THE MICROSCOPIC APPROACH TO $\mathcal{N} = 1$ SUPER YANG-MILLS THEORIES." International Journal of Modern Physics A 23, no. 14n15 (June 20, 2008): 2307–23. http://dx.doi.org/10.1142/s0217751x08041153.

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We give a brief account of the recent progresses in super Yang-Mills theories based in particular on the application of Nekrasov's instanton technology to the case of [Formula: see text] supersymmetry. We have developed a first-principle formalism from which any chiral observable in the theory can be computed, including in strongly coupled confining vacua. The correlators are first expressed in terms of some external variables as sums over colored partitions. The external variables are then fixed to their physical values by extremizing the microscopic quantum superpotential. Remarquably, the results can be shown to coincide with the Dijkgraaf-Vafa matrix model approach, which uses a totally different mathematical framework. These results clarify many important properties of [Formula: see text] theories, related in particular to generalized Konishi anomaly equations and to Veneziano-Yankielowicz terms in the glueball superpotentials. The proof of the equivalence between the formalisms based on colored partitions and on matrices is also a proof of the open/closed string duality in the chiral sector of the theories.

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21

Chan, Hong-Mo, and Sheung Tsun Tsou. "The framed Standard Model (I) — A physics case for framing the Yang–Mills theory?" International Journal of Modern Physics A 30, no. 30 (October 28, 2015): 1530059. http://dx.doi.org/10.1142/s0217751x15300598.

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Introducing, in the underlying gauge theory of the Standard Model, the frame vectors in internal space as field variables (framons), in addition to the usual gauge boson and matter fermions fields, one obtains: the standard Higgs scalar as the framon in the electroweak sector; a global [Formula: see text] symmetry dual to colour to play the role of fermion generations. Renormalization via framon loops changes the orientation in generation space of the vacuum, hence also of the mass matrices of leptons and quarks, thus making them rotate with changing scale [Formula: see text]. From previous work, it is known already that a rotating mass matrix will lead automatically to: CKM mixing and neutrino oscillations, hierarchical masses for quarks and leptons, a solution to the strong-CP problem transforming the theta-angle into a Kobayashi–Maskawa phase. Here in the framed standard model (FSM), the renormalization group equation has some special properties which explain the main qualitative features seen in experiment both for mixing matrices of quarks and leptons, and for their mass spectrum. Quantitative results will be given in Paper II. The present paper ends with some tentative predictions on Higgs decay, and with some speculations on the origin of dark matter.

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22

ANAGNOSTOPOULOS, K. N., W. BIETENHOLZ, and J. NISHIMURA. "THE AREA LAW IN MATRIX MODELS FOR LARGE N QCD STRINGS." International Journal of Modern Physics C 13, no. 04 (May 2002): 555–63. http://dx.doi.org/10.1142/s0129183102003334.

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We study the question whether matrix models obtained in the zero volume limit of 4d Yang–Mills theories can describe large N QCD strings. The matrix model we use is a variant of the Eguchi–Kawai model in terms of Hermitian matrices, but without any twists or quenching. This model was originally proposed as a toy model of the IIB matrix model. In contrast to common expectations, we do observe the area law for Wilson loops in a significant range of scale of the loop area. Numerical simulations show that this range is stable as N increases up to 768, which strongly suggests that it persists in the large N limit. Hence the equivalence to QCD strings may hold for length scales inside a finite regime.

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23

Castro, Carlos. "A Clifford algebra-based grand unification program of gravity and the Standard Model: a review study." Canadian Journal of Physics 92, no. 12 (December 2014): 1501–27. http://dx.doi.org/10.1139/cjp-2013-0686.

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A Clifford Cl(5, C) unified gauge field theory formulation of conformal gravity and U(4) × U(4) × U(4) Yang–Mills in 4D, is reviewed along with its implications for the Pati–Salam (PS) group SU(4) × SU(2)L × SU(2)R, and trinification grand unified theory models of three fermion generations based on the group SU(3)C × SU(3)L × SU(3)R. We proceed with a brief review of a unification program of 4D gravity and SU(3) × SU(2) × U(1) Yang–Mills emerging from 8D pure quaternionic gravity. A realization of E8 in terms of the Cl(16) = Cl(8) ⊗ Cl(8) generators follows, as a preamble to F. Smith’s E8 and Cl(16) = Cl(8) ⊗ Cl(8) unification model in 8D. The study of chiral fermions and instanton backgrounds in CP2 and CP3 related to the problem of obtaining three fermion generations is thoroughly studied. We continue with the evaluation of the coupling constants and particle masses based on the geometry of bounded complex homogeneous domains and geometric probability theory. An analysis of neutrino masses, Cabbibo–Kobayashi–Maskawa quark-mixing matrix parameters, and neutrino-mixing matrix parameters follows. We finalize with some concluding remarks about other proposals for the unification of gravity and the Standard Model, like string, M, and F theories and noncommutative and nonassociative geometry.

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24

SUGINO, FUMIHIKO. "COHOMOLOGICAL FIELD THEORY APPROACH TO MATRIX STRINGS." International Journal of Modern Physics A 14, no. 25 (October 10, 1999): 3979–4002. http://dx.doi.org/10.1142/s0217751x99001871.

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In this paper we consider IIA and IIB matrix string theories which are defined by two-dimensional and three-dimensional super Yang–Mills theory with the maximal supersymmetry, respectively. We exactly compute the partition function of both of the theories by mapping to a cohomological field theory. Our result for the IIA matrix string theory coincides with the result obtained in the infrared limit by Kostov and Vanhove, and thus gives a proof of the exact quasiclassics conjectured by them. Further, our result for the IIB matrix string theory coincides with the exact result of IKKT model by Moore, Nekrasov and Shatashvili. It may be an evidence of the equivalence between the two distinct IIB matrix models arising from different roots.

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25

STEINACKER, HAROLD. "NON-COMMUTATIVE GAUGE THEORY ON FUZZY ℂP2." Modern Physics Letters A 20, no. 17n18 (June 14, 2005): 1345–57. http://dx.doi.org/10.1142/s0217732305017809.

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Gauge theory on fuzzy ℂP2 can be defined as a multi-matrix model, which consistently combines a UV cutoff with the classical symmetries of ℂP2. The degrees of freedom are 8 hermitian matrices of finite size, 4 of which are tangential gauge fields and 4 are auxiliary variables. The model depends on a noncommutativity parameter [Formula: see text], and reduces to the usual U(n) Yang-Mills action on the 4-dimensional classical ℂP2 in the limit N→∞. The quantization of the model is defined in terms of a path integral, which is manifestly finite.

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26

Vizgin, Vladimir P. "“Comedy of mistakes” and “drama of humans”: on the domestic contribution to the creation of The Standard Model of elemantary particle in physics." Science management: theory and practice 2, no. 3 (2020): 196–224. http://dx.doi.org/10.19181/smtp.2020.2.3.11.

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The article explores domestic contribution to the creation of The Standard Model (SM). SM is a quantum field gauge theory of electromagnetic, weak and strong interactions, which is the basis of the modern theory of elementary particles. The process of its development covers a twenty-year period – from 1954 (the concept of non-Abelian Yang-Mills gauge fields) to the early 1970s, when the construction of renormalizable quantum chromodynamics and electroweak theory was completed. The reasons for the difficult perception of the Yang-Mills gauge field concept in the USSR are analyzed, associated primarily with the problem of “zero-charge” in quantum electrodynamics, and then in field theories of strong and weak interactions. This result, obtained by the leaders of the outstanding Russian scientific schools of theoretical physics, L. D. Landau, I. Ya. Pomeranchuk and their students, led to the rejection of the majority of Soviet physicists from field theory and to their transition to the position of a non-field phenomenological program (based on the S-matrix theory) in the construction of the theory of elementary particles.

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27

López-Picón, José Luis, Octavio Obregón, and José Ríos-Padilla. "A Proposal to Solve Finite N Matrix Theory: Reduced Model Related to Quantum Cosmology." Universe 8, no. 8 (August 11, 2022): 418. http://dx.doi.org/10.3390/universe8080418.

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The SU(N) invariant model of matrix theory that emerges as the regularization of the 11-dimensional super membrane is studied. This matrix model is identified with M theory in the limit N→∞. It has been conjectured that matrix models are also relevant for finite N where several examples and arguments have been given in the literature. By the use of a Dirac-like formulation usually developed in finding solutions in Supersymmetric Quantum Cosmology, we exhibit a method that could solve, in principle, any finite N model. As an example of our procedure, we choose a reduced SU(2) model and also show that this matrix model contains relevant supersymmetric quantum cosmological models as solutions. By these means, our solutions constitute an example in order to consider why the finite N matrix models are also relevant. Since the degrees of freedom of matrix models are, in some limit, identified with those of Super Yang Mills Theory SYM with a finite number of supercharges, our methodology offers the possibility, through some but yet unspecified identification, to relate the quantization presented here with that of SYM theory for any finite N.

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28

Varshovi, Amir Abbass. "⋆-cohomology, third type Chern character and anomalies in general translation-invariant noncommutative Yang–Mills." International Journal of Geometric Methods in Modern Physics 18, no. 06 (February 24, 2021): 2150089. http://dx.doi.org/10.1142/s0219887821500894.

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A representation of general translation-invariant star products ⋆ in the algebra of [Formula: see text] is introduced which results in the Moyal–Weyl–Wigner quantization. It provides a matrix model for general translation-invariant noncommutative quantum field theories in terms of the noncommutative calculus on differential graded algebras. Upon this machinery a cohomology theory, the so-called ⋆-cohomology, with groups [Formula: see text], [Formula: see text], is worked out which provides a cohomological framework to formulate general translation-invariant noncommutative quantum field theories based on the achievements for the commutative fields, and is comparable to the Seiberg–Witten map for the Moyal case. Employing the Chern–Weil theory via the integral classes of [Formula: see text] a noncommutative version of the Chern character is defined as an equivariant form which contains topological information about the corresponding translation-invariant noncommutative Yang–Mills theory. Thereby, we study the mentioned Yang–Mills theories with three types of actions of the gauge fields on the spinors, the ordinary, the inverse, and the adjoint action, and then some exact solutions for their anomalous behaviors are worked out via employing the homotopic correlation on the integral classes of ⋆-cohomology. Finally, the corresponding consistent anomalies are also derived from this topological Chern character in the ⋆-cohomology.

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29

ISHIKI, GORO. "MATRIX REGULARIZATION OF N = 4 SYM ON R × S3." International Journal of Modern Physics A 23, no. 14n15 (June 20, 2008): 2199–200. http://dx.doi.org/10.1142/s0217751x08040834.

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We revealed a relationship between the plane wave matrix model (PWMM) and N =4 super Yang-Mills (SYM) theory on R × S3: N =4 SYM on R × S3 is equivalent to the theory around a certain vacuum of PWMM. It is suggested from this relation that N =4 SYM on R × S3 is regularized by PWMM in the planar limit. Because PWMM originally possesses the gauge symmetry and SU(2|4) symmetry, this regularization also preserves these symmetries. In order to check the validity of this matrix regularization method, we calculate the Ward identity and the beta function at the 1-loop level. We find that the Ward identity is satisfied and the beta function vanishes in the continuum limit. The former result is consistent with the gauge symmetry of PWMM. The latter suggests the possibility that the conformal symmety is restored in the continuum limit.

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30

KRISHNASWAMI, GOVIND S. "2 + 1 ABELIAN "GAUGE THEORY" INSPIRED BY IDEAL HYDRODYNAMICS." International Journal of Modern Physics A 21, no. 18 (July 20, 2006): 3771–808. http://dx.doi.org/10.1142/s0217751x06030977.

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We study a possibly integrable model of Abelian gauge fields on a two-dimensional surface M, with volume form μ. It has the same phase-space as ideal hydrodynamics, a coadjoint orbit of the volume-preserving diffeomorphism group of M. Gauge field Poisson brackets differ from the Heisenberg algebra, but are reminiscent of Yang–Mills theory on a null surface. Enstrophy invariants are Casimirs of the Poisson algebra of gauge invariant observables. Some symplectic leaves of the Poisson manifold are identified. The Hamiltonian is a magnetic energy, similar to that of electrodynamics, and depends on a metric whose volume element is not a multiple of μ. The magnetic field evolves by a quadratically nonlinear "Euler" equation, which may also be regarded as describing geodesic flow on SDiff (M, μ). Static solutions are obtained. For uniform μ, an infinite sequence of local conserved charges beginning with the Hamiltonian are found. The charges are shown to be in involution, suggesting integrability. Besides being a theory of a novel kind of ideal flow, this is a toy-model for Yang–Mills theory and matrix field theories, whose gauge-invariant phase-space is conjectured to be a coadjoint orbit of the diffeomorphism group of a noncommutative space.

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31

Singh, Tejinder P. "Octonions, trace dynamics and noncommutative geometry—A case for unification in spontaneous quantum gravity." Zeitschrift für Naturforschung A 75, no. 12 (November 18, 2020): 1051–62. http://dx.doi.org/10.1515/zna-2020-0196.

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AbstractWe have recently proposed a new matrix dynamics at the Planck scale, building on the theory of trace dynamics and Connes noncommutative geometry program. This is a Lagrangian dynamics in which the matrix degrees of freedom are made from Grassmann numbers, and the Lagrangian is trace of a matrix polynomial. Matrices made from even grade elements of the Grassmann algebra are called bosonic, and those made from odd grade elements are called fermionic—together they describe an ‘aikyon’. The Lagrangian of the theory is invariant under global unitary transformations and describes gravity and Yang–Mills fields coupled to fermions. In the present article, we provide a basic definition of spin angular momentum in this matrix dynamics and introduce a bosonic(fermionic) configuration variable conjugate to the spin of a boson(fermion). We then show that at energies below Planck scale, where the matrix dynamics reduces to quantum theory, fermions have half-integer spin (in multiples of Planck’s constant), and bosons have integral spin. We also show that this definition of spin agrees with the conventional understanding of spin in relativistic quantum mechanics. Consequently, we obtain an elementary proof for the spin-statistics connection. We then motivate why an octonionic space is the natural space in which an aikyon evolves. The group of automorphisms in this space is the exceptional Lie group G2 which has 14 generators [could they stand for the 12 vector bosons and two degrees of freedom of the graviton?]. The aikyon also resembles a closed string, and it has been suggested in the literature that 10-D string theory can be represented as a 2-D string in the 8-D octonionic space. From the work of Cohl Furey and others it is known that the Dixon algebra made from the four division algebras [real numbers, complex numbers, quaternions and octonions] can possibly describe the symmetries of the standard model. In the present paper we outline how in our work the Dixon algebra arises naturally and could lead to a unification of gravity with the standard model. From this matrix dynamics, local quantum field theory arises as a low energy limit of this Planck scale dynamics of aikyons, and classical general relativity arises as a consequence of spontaneous localisation of a large number of entangled aikyons. We propose that classical curved space–time and Yang–Mills fields arise from an effective gauging which results from the collection of symmetry groups of the spontaneously localised fermions. Our work suggests that we live in an eight-dimensional octonionic universe, four of these dimensions constitute space–time and the other four constitute the octonionic internal directions on which the standard model forces live.

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32

Pawelczyk, Jacek. "Matrix models for Yang-Mills interactions." Journal of High Energy Physics 2000, no. 02 (February 22, 2000): 038. http://dx.doi.org/10.1088/1126-6708/2000/02/038.

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Austing, Peter, John F. Wheater, and Graziano Vernizzi. "Polyakov lines in Yang-Mills matrix models." Journal of High Energy Physics 2003, no. 09 (September 9, 2003): 023. http://dx.doi.org/10.1088/1126-6708/2003/09/023.

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34

Minahan, Joseph A. "Matrix models for 5d super Yang–Mills." Journal of Physics A: Mathematical and Theoretical 50, no. 44 (October 13, 2017): 443015. http://dx.doi.org/10.1088/1751-8121/aa5cbe.

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35

Ferrari, Frank. "Super Yang–Mills, matrix models and geometric transitions." Comptes Rendus Physique 6, no. 2 (March 2005): 219–30. http://dx.doi.org/10.1016/j.crhy.2004.12.002.

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36

Ydri, Badis. "Remarks on the eigenvalues distributions of D≤4 Yang–Mills matrix models." International Journal of Modern Physics A 30, no. 01 (January 9, 2015): 1450197. http://dx.doi.org/10.1142/s0217751x14501978.

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The phenomenon of emergent fuzzy geometry and noncommutative gauge theory from Yang–Mills matrix models is briefly reviewed. In particular, the eigenvalue distributions of Yang–Mills matrix models in lower dimensions in the commuting (matrix or Yang–Mills) phase of these models are discussed.

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37

Oktay, O. "Thermalization in Massive Deformations of Yang–Mills Matrix Models." Acta Physica Polonica B 52, no. 12 (2021): 1405. http://dx.doi.org/10.5506/aphyspolb.52.1405.

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38

Steinacker, Harold C. "Quantized open FRW cosmology from Yang–Mills matrix models." Physics Letters B 782 (July 2018): 176–80. http://dx.doi.org/10.1016/j.physletb.2018.05.011.

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39

Steinacker, Harold. "Emergent gravity and noncommutative branes from Yang–Mills matrix models." Nuclear Physics B 810, no. 1-2 (March 2009): 1–39. http://dx.doi.org/10.1016/j.nuclphysb.2008.10.014.

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40

O’Connor, D. "Low-dimensional Yang-Mills theories: Matrix models and emergent geometry." Theoretical and Mathematical Physics 169, no. 1 (October 2011): 1405–12. http://dx.doi.org/10.1007/s11232-011-0116-9.

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41

Bellucci, Stefano, and Corneliu Sochichiu. "On matrix models for anomalous dimensions of super-Yang–Mills theory." Nuclear Physics B 726, no. 1-2 (October 2005): 233–51. http://dx.doi.org/10.1016/j.nuclphysb.2005.07.026.

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42

YDRI, BADIS. "IMPACT OF SUPERSYMMETRY ON EMERGENT GEOMETRY IN YANG–MILLS MATRIX MODELS II." International Journal of Modern Physics A 27, no. 17 (June 26, 2012): 1250088. http://dx.doi.org/10.1142/s0217751x12500881.

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We present a study of D = 4 supersymmetric Yang–Mills matrix models with SO(3) mass terms based on the Monte Carlo method. In the bosonic models we show the existence of an exotic first-/second-order transition from a phase with a well defined background geometry (the fuzzy sphere) to a phase with commuting matrices with no geometry in the sense of Connes. At the transition point the sphere expands abruptly to infinite size then it evaporates as we increase the temperature (the gauge coupling constant). The transition looks first-order due to the discontinuity in the action whereas it looks second-order due to the divergent peak in the specific heat. The fuzzy sphere is stable for the supersymmetric models in the sense that the bosonic phase transition is turned into a very slow crossover transition. The transition point is found to scale to zero with N. We conjecture that the transition from the background sphere to the phase of commuting matrices is associated with spontaneous supersymmetry breaking. The eigenvalues distribution of any of the bosonic matrices in the matrix phase is found to be given by a nonpolynomial law obtained from the fact that the joint probability distribution of the four matrices is uniform inside a solid ball with radius R. The eigenvalues of the gauge field on the background geometry are also found to be distributed according to this nonpolynomial law.

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Steinacker, Harold. "Covariant field equations, gauge fields and conservation laws from Yang-Mills matrix models." Journal of High Energy Physics 2009, no. 02 (February 17, 2009): 044. http://dx.doi.org/10.1088/1126-6708/2009/02/044.

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Sperling, Marcus, and Harold C. Steinacker. "The fuzzy 4-hyperboloid Hn4 and higher-spin in Yang–Mills matrix models." Nuclear Physics B 941 (April 2019): 680–743. http://dx.doi.org/10.1016/j.nuclphysb.2019.02.027.

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STEINACKER, HAROLD. "EMERGENT GRAVITY AND GAUGE THEORY FROM MATRIX MODELS." International Journal of Modern Physics A 24, no. 15 (June 20, 2009): 2866–76. http://dx.doi.org/10.1142/s0217751x09046217.

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Matrix models of Yang-Mills type induce an effective gravity theory on 4-dimensional branes, which are considered as models for dynamical space-time. We review recent progress in the understanding of this emergent gravity. The metric is not fundamental but arises effectively in the semi-classical limit, along with nonabelian gauge fields. This leads to a mechanism which could resolve the cosmological constant problem.

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Steinacker, Harold C. "Higher-spin kinematics & no ghosts on quantum space-time in Yang–Mills matrix models." Advances in Theoretical and Mathematical Physics 25, no. 4 (2021): 1025–93. http://dx.doi.org/10.4310/atmp.2021.v25.n4.a4.

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47

Krishnaswami, Govind S. "Schwinger–Dyson operator of Yang–Mills matrix models with ghosts and derivations of the graded shuffle algebra." Journal of Physics A: Mathematical and Theoretical 41, no. 14 (March 26, 2008): 145402. http://dx.doi.org/10.1088/1751-8113/41/14/145402.

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48

Steinacker, Harold C. "Scalar modes and the linearized Schwarzschild solution on a quantized FLRW space-time in Yang–Mills matrix models." Classical and Quantum Gravity 36, no. 20 (September 18, 2019): 205005. http://dx.doi.org/10.1088/1361-6382/ab39e3.

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49

LEE, C. W. H., and S. G. RAJEEV. "A REVIEW OF SYMMETRY ALGEBRAS OF QUANTUM MATRIX MODELS IN THE LARGE N LIMIT." International Journal of Modern Physics A 14, no. 28 (November 10, 1999): 4395–455. http://dx.doi.org/10.1142/s0217751x99002074.

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This is a review article in which we will introduce, in a unifying fashion and with more intermediate steps in some difficult calculations, two infinite-dimensional Lie algebras of quantum matrix models, one for the open string sector and the other for the closed string sector. Physical observables of quantum matrix models in the large N limit can be expressed as elements of these Lie algebras. We will see that both algebras arise as quotient algebras of a larger Lie algebra. We will also discuss some properties of these Lie algebras not published elsewhere yet, and briefly review their relationship with well-known algebras like the Cuntz algebra, the Witt algebra and the Virasoro algebra. We will also review how the Yang–Mills theory, various low energy effective models of string theory, quantum gravity, string-bit models, and the quantum spin chain models can be formulated as quantum matrix models. Studying these algebras thus help us understand the common symmetry of these physical systems.

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50

Reshetnyak, Alexander. "On gauge independence for gauge models with soft breaking of BRST symmetry." International Journal of Modern Physics A 29, no. 30 (December 8, 2014): 1450184. http://dx.doi.org/10.1142/s0217751x1450184x.

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A consistent quantum treatment of general gauge theories with an arbitrary gauge-fixing in the presence of soft breaking of the BRST symmetry in the field–antifield formalism is developed. It is based on a gauged (involving a field-dependent parameter) version of finite BRST transformations. The prescription allows one to restore the gauge-independence of the effective action at its extremals and therefore also that of the conventional S-matrix for a theory with BRST-breaking terms being additively introduced into a BRST-invariant action in order to achieve a consistency of the functional integral. We demonstrate the applicability of this prescription within the approach of functional renormalization group to the Yang–Mills and gravity theories. The Gribov–Zwanziger action and the refined Gribov–Zwanziger action for a many-parameter family of gauges, including the Coulomb, axial and covariant gauges, are derived perturbatively on the basis of finite gauged BRST transformations starting from Landau gauge. It is proved that gauge theories with soft breaking of BRST symmetry can be made consistent if the transformed BRST-breaking terms satisfy the same soft BRST symmetry breaking condition in the resulting gauge as the untransformed ones in the initial gauge, and also without this requirement.

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