Equivalence Postulate and the Quantum Potential of Two Free Particles

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.Comment: 7 pages, LaTeX. Talk at the First International Conference on String Cosmology. Oxford, United Kingdom. July 200

Equivalence Postulate and Quantum Origin of Gravitation

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.Comment: 1+19 pages, minor changes and typos corrected, to appear in Found. Phys. Let

Quantum Field Perturbation Theory Revisited

Schwinger's formalism in quantum field theory can be easily implemented in the case of scalar theories in $D$ dimension with exponential interactions, such as $\mu^D\exp(\alpha\phi)$. In particular, we use the relation $$\exp\big(\alpha{\delta\over \delta J(x)}\big)\exp(-Z_0[J])=\exp(-Z_0[J+\alpha_x])$$ with $J$ the external source, and $\alpha_x(y)=\alpha\delta(y-x)$. Such a shift is strictly related to the normal ordering of $\exp(\alpha\phi)$ and to a scaling relation which follows by renormalizing $\mu$. Next, we derive a new formulation of perturbation theory for the potentials $V(\phi)={\lambda\over n!}:\phi^n:$, using the generating functional associated to $:\exp(\alpha\phi):$. The $\Delta(0)$-terms related to the normal ordering are absorbed at once. The functional derivatives with respect to $J$ to compute the generating functional are replaced by ordinary derivatives with respect to auxiliary parameters. We focus on scalar theories, but the method is general and similar investigations extend to other theories.Comment: 21 pages. Includes a modified Feynman propagator which is massless in D=4 and scaling relations for the generating functional. References added. PRD versio

A Surprising Relation for the Effective Coupling Constants of N=2 Super Yang-Mills Theories

We show that the effective coupling constants $\tau$ of supersymmetric gauge theories described by hyperelliptic curves do not distinguish between the lattices of the two kinds of heterotic string. In particular, the following relation $$\Theta_{D_{16}^+}(\tau)=\Theta_{E_8}^2(\tau)$$ holds. This is reminiscent of the relation, by $T$-duality, of the two heterotic strings. We suggest that such a relation extends to all curves describing effective supersymmetric gauge theories.Comment: 7 page

Modular Invariant Regularization of String Determinants and the Serre GAGA Principle

Since any string theory involves a path integration on the world-sheet metric, their partition functions are volume forms on the moduli space of genus g Riemann surfaces M_g, or on its super analog. It is well known that modular invariance fixes strong constraints that in some cases appear only at higher genus. Here we classify all the Weyl and modular invariant partition functions given by the path integral on the world-sheet metric, together with space-time coordinates, b-c and/or beta-gamma systems, that correspond to volume forms on M_g. This was a long standing question, advocated by Belavin and Knizhnik, inspired by the Serre GAGA principle and based on the properties of the Mumford forms. The key observation is that the Bergman reproducing kernel provides a Weyl and modular invariant way to remove the point dependence that appears in the above string determinants, a property that should have its superanalog based on the super Bergman reproducing kernel. This is strictly related to the properties of the propagator associated to the space-time coordinates. Such partition functions Z[J] have well-defined asymptotic behavior and can be considered as a basis to represent a wide class of string theories. In particular, since non-critical bosonic string partition functions Z_D are volume forms on M_g, we suggest that there is a mapping, based on bosonization and degeneration techniques, from the Liouville sector to first order systems that may identify Z_D as a subclass of the Z[J]. The appearance of b-c and beta-gamma systems of any conformal weight shows that such theories are related to W algebras. The fact that in a large N 't Hooft-like limit 2D W_N minimal models CFTs are related to higher spin gravitational theories on AdS_3, suggests that the string partition functions introduced here may lead to a formulation of higher spin theories in a string context.Comment: 30 pp. New results on Quillen metric and its relations with the Mumford forms and the tautological classes. References adde

"Thermodynamique cach\'ee des particules" and the quantum potential

According to de Broglie, temperature plays a basic role in quantum Hamilton-Jacobi theory. Here we show that a possible dependence on the temperature of the integration constants of the relativistic quantum Hamilton-Jacobi may lead to corrections to the dispersion relations. The change of the relativistic equations is simply described by means of a thermal coordinate.Comment: 8 page

Classification of Commutator Algebras Leading to the New Type of Closed Baker-Campbell-Hausdorff Formulas

We show that there are {\it 13 types} of commutator algebras leading to the new closed forms of the Baker-Campbell-Hausdorff (BCH) formula $$\exp(X)\exp(Y)\exp(Z)=\exp({AX+BZ+CY+DI}) \ ,$$ derived in arXiv:1502.06589, JHEP {\bf 1505} (2015) 113. This includes, as a particular case, $\exp(X) \exp(Z)$, with $[X,Z]$ containing other elements in addition to $X$ and $Z$. The algorithm exploits the associativity of the BCH formula and is based on the decomposition $\exp(X)\exp(Y)\exp(Z)=\exp(X)\exp({\alpha Y}) \exp({(1-\alpha) Y}) \exp(Z)$, with $\alpha$ fixed in such a way that it reduces to $\exp({\tilde X})\exp({\tilde Y})$, with $\tilde X$ and $\tilde Y$ satisfying the Van-Brunt and Visser condition $[\tilde X,\tilde Y]=\tilde u\tilde X+\tilde v\tilde Y+\tilde cI$. It turns out that $e^\alpha$ satisfies, in the generic case, an algebraic equation whose exponents depend on the parameters defining the commutator algebra. In nine {\it types} of commutator algebras, such an equation leads to rational solutions for $\alpha$. We find all the equations that characterize the solution of the above decomposition problem by combining it with the Jacobi identity.Comment: 15 pages. Typos corrected. JGP versio

Modular Invariance and Structure of the Exact Wilsonian Action of N=2 SYM

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.Comment: 10 pages, LaTeX file, misprints and a factor 2 in the derivation of alpha correcte

Vector-Valued Modular Forms from the Mumford Form, Schottky-Igusa Form, Product of Thetanullwerte and the Amazing Klein Formula

Vector-valued Siegel modular forms are the natural generalization of the classical elliptic modular forms as seen by studying the cohomology of the universal abelian variety. We show that for g>=4, a new class of vector-valued modular forms, defined on the Teichmuller space, naturally appears from the Mumford forms, a question directly related to the Schottky problem. In this framework we show that the discriminant of the quadric associated to the complex curves of genus 4 is proportional to the square root of the products of Thetanullwerte \chi_{68}, which is a proof of the recently rediscovered Klein `amazing formula'. Furthermore, it turns out that the coefficients of such a quadric are derivatives of the Schottky-Igusa form evaluated at the Jacobian locus, implying new theta relations involving the latter, \chi_{68} and the theta series corresponding to the even unimodular lattices E_8\oplus E_8 and D_{16}^+. We also find, for g=4, a functional relation between the singular component of the theta divisor and the Riemann period matrix.Comment: 17 pages. Final version in Proc. Amer. Math. So

The Singular Locus of the Theta Divisor and Quadrics through a Canonical Curve

A section K on a genus g canonical curve C is identified as the key tool to prove new results on the geometry of the singular locus Theta_s of the theta divisor. The K divisor is characterized by the condition of linear dependence of a set of quadrics containing C and naturally associated to a degree g effective divisor on C. K counts the number of intersections of special varieties on the Jacobian torus defined in terms of Theta_s. It also identifies sections of line bundles on the moduli space of algebraic curves, closely related to the Mumford isomorphism, whose zero loci characterize special varieties in the framework of the Andreotti-Mayer approach to the Schottky problem, a result which also reproduces the only previously known case g=4. This new approach, based on the combinatorics of determinantal relations for two-fold products of holomorphic abelian differentials, sheds light on basic structures, and leads to the explicit expressions, in terms of theta functions, of the canonical basis of the abelian holomorphic differentials and of the constant defining the Mumford form. Furthermore, the metric on the moduli space of canonical curves, induced by the Siegel metric, which is shown to be equivalent to the Kodaira-Spencer map of the square of the Bergman reproducing kernel, is explicitly expressed in terms of the Riemann period matrix only, a result previously known for the trivial cases g=2 and g=3. Finally, the induced Siegel volume form is expressed in terms of the Mumford form.Comment: 88+1 page