[Paper Review] From the DeDonder-Weyl Hamiltonian formalism to quantization of Gravity

Igor V. Kanatchikov|ArXiv.org|Oct 22, 1998

Noncommutative and Quantum Gravity Theories6 references18 citations

TL;DR

This paper proposes a manifestly covariant quantization framework for gravity and fields using the De Donder-Weyl (DW) Hamiltonian formalism, replacing complex numbers with spacetime Clifford algebra to yield a hypercomplex quantum theory. It formulates a covariant hypercomplex Schrödinger equation and sketches its application to General Relativity, offering a time-independent, geometrically unified approach to quantum gravity that avoids the conventional problem of time.

ABSTRACT

An approach to quantization of fields and gravity based on the De Donder-Weyl covariant Hamiltonian formalism is outlined. It leads to a hypercomplex extension of quantum mechanics in which the algebra of complex numbers is replaced by the space-time Clifford algebra and all space-time variables enter on equal footing. A covariant hypercomplex analogue of the Schrödinger equation is formulated. Elements of quantization of General Relativity within the present framework are sketched.

Motivation & Objective

• To address the 'problem of time' in quantum gravity by avoiding the need to single out a time coordinate in the Hamiltonian formulation.
• To develop a manifestly covariant quantization scheme for field theories and gravity using the De Donder-Weyl (DW) Hamiltonian formalism.
• To generalize quantum mechanics by replacing complex numbers with spacetime Clifford algebra, leading to a hypercomplex quantum theory.
• To formulate a covariant hypercomplex analogue of the Schrödinger equation applicable to relativistic field theories and gravity.
• To explore the viability of the Dirac-Kähler equation as a universal wave function formalism in curved spacetime, avoiding spinor structure restrictions.

Proposed method

• Utilizes the De Donder-Weyl Hamiltonian formalism, defining polymomenta $ p_a^\mu = \partial L / \partial(\partial_\mu y^a) $ and DW Hamiltonian $ H = \partial_\mu y^a p_a^\mu - L $, to derive covariant field equations.
• Applies Poisson brackets on differential forms (0- and (n-1)-forms) to identify canonically conjugate variables and generalize canonical commutation relations.
• Constructs a hypercomplex quantum theory by replacing complex numbers with elements of the spacetime Clifford algebra, leading to a covariant hypercomplex Schrödinger equation.
• Derives a generalized Schrödinger equation in curved spacetime using the Dirac-Kähler operator $ \hat{\not{D}}_\Gamma = d_\Gamma - \delta_\Gamma $, with wave function as a non-homogeneous differential form.
• Represents operators via Atiyah-Kähler algebra elements (e.g., $ \partial_\mu \hskip 2.0pt\text{\textasciibreve{}} $ and $ dx^\mu \wedge $) rather than Dirac matrices, enabling metric-based quantization without tetrads.
• Proposes a generalized Schrödinger equation for gravity: $ i\hbar\kappa\sqrt{|g|}\,\widehat{\not{D}}_\Gamma \Psi = \widehat{H}\Psi $, with $ \Psi $ as a Clifford-algebra-valued non-homogeneous form.

Experimental results

Research questions

• RQ1 Can a manifestly covariant quantization scheme for gravity be constructed without privileging a time variable?
• RQ2 How can the standard complex-number-based Schrödinger equation be generalized to a spacetime-covariant, hypercomplex form using Clifford algebra?
• RQ3 Is the Dirac-Kähler equation a viable universal wave function formalism in curved spacetime, independent of spinor structure?
• RQ4 Can the hypercomplex Schrödinger equation accommodate fields of different spins, including scalar, spinor, and vector fields?
• RQ5 How can operator ordering ambiguities in the quantized DW formalism be resolved while preserving covariance and consistency with the correspondence principle?

Key findings

• The De Donder-Weyl formalism enables a manifestly covariant Hamiltonian formulation of field theory, avoiding the need to foliate spacetime into space-like hypersurfaces.
• A Poisson bracket structure on differential forms (0- and (n-1)-forms) is established, generalizing canonical commutation relations to a covariant setting.
• The hypercomplex Schrödinger equation $ i\hbar \kappa \sqrt{|g|} \widehat{\not{D}}_\Gamma \Psi = \widehat{H} \Psi $ is proposed as a candidate for quantum gravity, with $ \Psi $ as a non-homogeneous differential form in the Atiyah-Kähler algebra.
• The framework allows quantization of metric gravity directly without requiring tetrad formalism, using only the spacetime metric and its connection.
• The Dirac-Kähler equation is suggested as a more fundamental alternative to the Dirac equation in curved spacetime, avoiding restrictions to spin-structure manifolds.
• The wave function $ \Psi $, when interpreted as a Clifford-algebra-valued form, may unify the description of fields of various spins, though its probabilistic interpretation remains an open issue.