arXiv subjects
Publications and source records attributed to Marco Matone.
We consider the chiral ring of the pure N=1 supersymmetric gauge theory with SU(N) gauge group and show that the classical relation S^{N^2}=0 is modified to the exact quantum relation (S^N-Λ^{3N})^N=0.
In the framework of the matrix model/gauge theory correspondence, we consider supersymmetric U(N) gauge theory with $U(1)^N$ symmetry breaking pattern. Due to the presence of the Veneziano--Yankielowicz effective superpotential, in order to satisfy the $F$--term condition $\sum_iS_i=0$, we are forced to introduce additional terms in the free energy of the corresponding matrix model with respect to the usual formulation. This leads to a matrix model formulation with a cubic potential which is free of parameters and displays a branched structure. In this way we naturally solve the usual problem of the identification between dimensionful and dimensionless quantities. Furthermore, we need not introduce the $\N=1$ scale by hand in the matrix model. These facts are related to remarkable coincidences which arise at the critical point and lead to a branched bare coupling constant. The latter plays the role of the $\N=1$ and $\N=2$ scale tuning parameter. We then show that a suitable rescaling leads to the correct identification of the $\N=2$ variables. Finally, by means of the the mentioned coincidences, we provide a direct expression for the $\N=2$ prepotential, including the gravitational corrections, in terms of the free energy. This suggests that the matrix model provides a triangulation of the istanton moduli space.
We show that a suitable rescaling of the matrix model coupling constant makes manifest the duality group of the N=2 SYM theory with gauge group SU(2). This is done by first identifying the possible modifications of the SYM moduli preserving the monodromy group. Then we show that in matrix models there is a simple rescaling of the pair $(S_D,S)$ which makes them dual variables with $Γ(2)$ monodromy. We then show that, thanks to a crucial scaling property of the free energy derived perturbatively by Dijkgraaf, Gukov, Kazakov and Vafa, this redefinition corresponds to a rescaling of the free energy which in turn fixes the rescaling of the coupling constant. Next, we show that in terms of the rescaled free energy one obtains a nonperturbative relation which is the matrix model counterpart of the relation between the $u$--modulus and the prepotential of N=2 SYM. This suggests considering a dual formulation of the matrix model in which the expansion of the prepotential in the strong coupling region, whose QFT derivation is still unknown, should follow from perturbation theory. The investigation concerns the SU(2) gauge group and can be generalized to higher rank groups.
Commutativity of the diagram of the maps connecting three one--particle state, implied by the Equivalence Postulate (EP), gives a cocycle condition which unequivocally leads to the quantum Hamilton--Jacobi equation. Energy quantization is a direct consequences of the local homeomorphicity of the trivializing map. We review the EP and show that the quantum potential for two free particles, which depends on constants which may have a geometrical interpretation, plays the role of interaction term that admits solutions which do not vanish in the classical limit.
We show that the pure spinor formalism proposed by Berkovits to covariantly quantize superstrings is a gauge fixed, twisted version of the complexified n=2 superembedding formulation of the superstring. This provides the Berkovits approach with a geometrical superdiffeomorphism invariant ground. As a consequence, the absence of the worldsheet (super)diffeomorphism ghosts in the pure spinor quantization prescription and the nature of the Berkovits BRST charge and antighost are clarified. Since superembedding is classically equivalent to the Green-Schwarz formulation, we thus also relate the latter to the pure spinor construction.
We suggest that quantum mechanics and gravity are intimately related. In particular, we investigate the quantum Hamilton-Jacobi equation in the case of two free particles and show that the quantum potential, which is attractive, may generate the gravitational potential. The investigation, related to the formulation of quantum mechanics based on the equivalence postulate, is based on the analysis of the reduced action. A consequence of this approach is that the quantum potential is always non-trivial even in the case of the free particle. It plays the role of intrinsic energy and may in fact be at the origin of fundamental interactions. We pursue this idea, by making a preliminary investigation of whether there exists a set of solutions for which the quantum potential can be expressed with a gravitational potential leading term which alone would remain in the limit hbar \to 0. A number of questions are raised for further investigation.
We use the recursive structure of the compactification of the instanton moduli space of N=2 Super Yang-Mills theory with gauge group SU(2), to construct, by inductive limit, a universal moduli space which includes all the multi-instanton moduli spaces. Furthermore, with the aim of understanding the field theoretic structure of the strong coupling expansion, we perform the Borel sum which acts on the parameter defining such a universal moduli space.
By using the recursion relations found in the framework of N=2 Super Yang-Mills theory with gauge group SU(2), we reconstruct the structure of the instanton moduli space and its volume form for all winding numbers. The construction is reminiscent of the Deligne-Knudsen-Mumford compactification and uses an analogue of the Wolpert restriction phenomenon which arises in the case of moduli spaces of Riemann surfaces.
We show that the recently proposed formulation of noncommutative N=2 Super Yang-Mills theory implies that the commutative and noncommutative effective coupling constants τ(u) and τ_{nc}(u) coincide. We then introduce a key relation which allows to find a nontrivial solution of such equation, thus fixing the form of the low-energy effective action. The dependence on the noncommutative parameter arises from a rational function deforming the Seiberg-Witten differential.
We show that the recently formulated Equivalence Principle (EP) implies a basic cocycle condition both in Euclidean and Minkowski spaces, which holds in any dimension. This condition, that in one-dimension is sufficient to fix the Schwarzian equation [6], implies a fundamental higher dimensional Moebius invariance which in turn univocally fixes the quantum version of the Hamilton-Jacobi equation. This holds also in the relativistic case, so that we obtain both the time-dependent Schroedinger equation and the Klein-Gordon equation in any dimension. We then show that the EP implies that masses are related by maps induced by the coordinate transformations connecting different physical systems. Furthermore, we show that the minimal coupling prescription, and therefore gauge invariance, arises quite naturally in implementing the EP. Finally, we show that there is an antisymmetric two-tensor which underlies Quantum Mechanics and sheds new light on the nature of the Quantum Hamilton-Jacobi equation.
Recently we showed that the postulated diffeomorphic equivalence of states implies quantum mechanics. This approach takes the canonical variables to be dependent by the relation p=\partial_q S_0 and exploits a basic GL(2,C)-symmetry which underlies the canonical formalism. In particular, we looked for the special transformations leading to the free system with vanishing energy. Furthermore, we saw that while on the one hand the equivalence principle cannot be consistently implemented in classical mechanics, on the other it naturally led to the quantum analogue of the Hamilton-Jacobi equation, thus implying the Schroedinger equation. In this letter we show that actually the principle uniquely leads to this solution. Furthermore, we find the map reducing any system to the free one with vanishing energy and derive the transformations on S_0 leaving the wave function invariant. We also express the canonical and Schroedinger equations by means of the brackets recently introduced in the framework of N=2 SYM. These brackets are the analogue of the Poisson brackets with the canonical variables taken as dependent.
The Equivalence Principle (EP), stating that all physical systems are connected by a coordinate transformation to the free one with vanishing energy, univocally leads to the Quantum Stationary HJ Equation (QSHJE). Trajectories depend on the Planck length through hidden variables which arise as initial conditions. The formulation has manifest p-q duality, a consequence of the involutive nature of the Legendre transform and of its recently observed relation with second-order linear differential equations. This reflects in an intrinsic psi^D-psi duality between linearly independent solutions of the Schroedinger equation. Unlike Bohm's theory, there is a non-trivial action even for bound states. No use of any axiomatic interpretation of the wave-function is made. Tunnelling is a direct consequence of the quantum potential which differs from the usual one and plays the role of particle's self-energy. The QSHJE is defined only if the ratio psi^D/psi is a local self-homeomorphism of the extended real line. This is an important feature as the L^2 condition, which in the usual formulation is a consequence of the axiomatic interpretation of the wave-function, directly follows as a basic theorem which only uses the geometrical gluing conditions of psi^D/psi at q=\pm\infty as implied by the EP. As a result, the EP itself implies a dynamical equation that does not require any further assumption and reproduces both tunnelling and energy quantization. Several features of the formulation show how the Copenhagen interpretation hides the underlying nature of QM. Finally, the non-stationary higher dimensional quantum HJ equation and the relativistic extension are derived.
We postulate that physical states are equivalent under coordinate transformations. We then implement this equivalence principle first in the case of one-dimensional stationary systems showing that it leads to the quantum analogue of the Hamilton-Jacobi equation which in turn implies the Schroedinger equation. In this context the Planck constant plays the role of covariantizing parameter. The construction is deeply related to the GL(2,C)-symmetry of the second-order differential equation associated to the Legendre transformation which selects, in the case of the quantum analogue of the Hamiltonian characteristic function, self-dual states which guarantee its existence for any physical system. The universal nature of the self-dual states implies the Schroedinger equation in any dimension.
The Quantum Stationary HJ Equation (QSHJE) that we derived from the equivalence principle, gives rise to initial conditions which cannot be seen in the Schroedinger equation. Existence of the classical limit leads to a dependence of the integration constant $\ell=\ell_1+i\ell_2$ on the Planck length. Solutions of the QSHJE provide a trajectory representation of quantum mechanics which, unlike Bohm's theory, has a non-trivial action even for bound states and no wave guide is present. The quantum potential turns out to be an intrinsic potential energy of the particle which, similarly to the relativistic rest energy, is never vanishing.
A basic aspect of the recently proposed approach to quantum mechanics is that no use of any axiomatic interpretation of the wave function is made. In particular, the quantum potential turns out to be an intrinsic potential energy of the particle, which, similarly to the relativistic rest energy, is never vanishing. This is related to the tunnel effect, a consequence of the fact that the conjugate momentum field is real even in the classically forbidden regions. The quantum stationary Hamilton-Jacobi equation is defined only if the ratio psi^D/psi of two real linearly independent solutions of the Schroedinger equation, and therefore of the trivializing map, is a local homeomorphism of the extended real line into itself, a consequence of the Moebius symmetry of the Schwarzian derivative. In this respect we prove a basic theorem relating the request of continuity at spatial infinity of psi^D/psi, a consequence of the q - 1/q duality of the Schwarzian derivative, to the existence of L^2(R) solutions of the corresponding Schroedinger equation. As a result, while in the conventional approach one needs the Schroedinger equation with the L^2(R) condition, consequence of the axiomatic interpretation of the wave function, the equivalence principle by itself implies a dynamical equation that does not need any assumption and reproduces both the tunnel effect and energy quantization.
We show that the stationary quantum Hamilton-Jacobi equation of non-relativistic 1D systems, underlying Bohmian mechanics, takes the classical form with $\partial_q$ replaced by $\partial_{\hat q}$ where $d\hat q={dq\over \sqrt{1-β^2}}$. The $β^2$ term essentially coincides with the quantum potential that, like $V-E$, turns out to be proportional to a curvature arising in projective geometry. In agreement with the recently formulated equivalence principle, these ``quantum transformations'' indicate that the classical and quantum potentials deform space geometry.
In this paper we first investigate the Ansatz of one of the present authors for K(Ψ,\barΨ), the adimensional modular invariant non-holomorphic correction to the Wilsonian effective Lagrangian of an N=2 globally supersymmetric gauge theory. The renormalisation group beta-function of the theory crucially allows us to express K(Ψ,\barΨ) in a form that easily generalises to the case in which the theory is coupled to N_F hypermultiplets in the fundamental representation of the gauge group. This function satisfies an equation which should be viewed as a fully non-perturbative ``non-chiral superconformal Ward identity". We also determine its renormalisation group equation. Furthermore, as a first step towards checking the validity of this Ansatz, we compute the contribution to K(Ψ,\barΨ) from instantons of winding number k=1 and k=2. As a by-product of our analysis we check a non-renormalisation theorem for N_F=4.
We construct modular invariants on the moduli space of quantum vacua of N=2 SYM with gauge group SU(2). We also introduce a nonchiral function K which is expressed in terms of the Seiberg-Witten and Poincare' metrics. It turns out that K has all the expected properties of the next to leading term in the Wilsonian effective action whose modular properties are considered in the framework of the dimensional regularization.