A New Look At The Path Integral Of Quantum Mechanics (1009.6032v1)

Abstract: The Feynman path integral of ordinary quantum mechanics is complexified and it is shown that possible integration cycles for this complexified integral are associated with branes in a two-dimensional A-model. This provides a fairly direct explanation of the relationship of the A-model to quantum mechanics; such a relationship has been explored from several points of view in the last few years. These phenomena have an analog for Chern-Simons gauge theory in three dimensions: integration cycles in the path integral of this theory can be derived from N=4 super Yang-Mills theory in four dimensions. Hence, under certain conditions, a Chern-Simons path integral in three dimensions is equivalent to an N=4 path integral in four dimensions.

• The paper introduces a complexification of Feynman path integrals that connects them with A-model branes in quantum mechanics.
• It links integration cycles from the complex approach to gauge theories, including Chern-Simons and super Yang-Mills models.
• It implies that alternative integration cycles can redefine quantization methods, opening new avenues in quantum field theory research.

A Formal Summary of "A New Look At The Path Integral Of Quantum Mechanics"

In this paper, the author explores the complexities of the Feynman path integral within quantum mechanics, presenting a novel approach by considering a complexification of the integral. The primary goal is to articulate the connection between the path integral formalism and various aspects of quantum field theory, particularly through the examination of branes in a two-dimensional $A$-model. This exploration has implications for related theories like the Chern-Simons gauge theory in three dimensions, which can, under certain conditions, be equated to a path integral in a four-dimensional $=4$ super Yang-Mills theory.

Key Concepts and Methodological Approach

1. Complexification of Path Integrals:
   • The author begins by introducing a complexified version of the Feynman path integral, which takes the form $\int\,D\Phi\,\exp\left(iI(\Phi)\right)$ in its Lorentzian signature. By transitioning to a Euclidean version, the author sets the stage for exploring integration cycles that are associated with certain types of branes in a two-dimensional $A$-model framework.
2. Relationship with the $A$-Model:
   • Through the complexification process, the path integral variables are connected to branes within the $A$-model, where the variables become boundary values of pseudoholomorphic maps. This relationship draws on previous work considering $A$-models and quantization, extending the understanding by proposing that coisotropic $A$-branes yield new integration cycles in quantum mechanics.
3. Applications to Gauge Theories:
   • The paper extends these ideas into the field of gauge theories, exemplified by the Chern-Simons gauge theory. Here, the integration cycles, derived from a higher-dimensional space, suggest that certain path integrals in three dimensions have equivalents in four-dimensional super Yang-Mills theory.
4. Integration Cycles and Quantization:
   • The author proposes that different integration cycles in the path integral correspond potentially to different quantum systems. This implies a strong link between the theoretical structure of quantum field theories in varying dimensions and suggests new approaches to quantization.

Numerical Results and Claims

The paper boldly suggests that the proposed integration cycles can transform the approach to quantization in quantum mechanics, potentially leading to refined understanding and applications in other quantum field theories. This theoretical perspective opens pathways for more detailed inquiry, particularly in the domain of non-relativistic quantum mechanics with pseudoholomorphic maps.

Theoretical and Practical Implications

The work lays a foundational basis for employing complexified variables and brane structures to yield insights into the quantum mechanical framework from a higher-dimensional perspective. The implications span both theoretical physics, by reinforcing connections between various dimensions and models, and practical applications concerning gauge theory analysis and quantization. This development showcases the applicability of rigorous mathematical structures to enhance and potentially redefine approaches in quantum physics.

Speculation on Future Developments

Looking forward, the author anticipates further exploration into the symbiotic relationship between field theories of differing dimensions. This includes deeper investigations into other gauge theories and potential applications in quantum gravity, string theory, and related disciplines where the path integral plays a critical role. Such advancements could lead to breakthroughs in understanding fundamental interactions and the nature of quantized spaces in physics.

This paper offers a technically robust and conceptually stimulating exploration, inviting future examination and validation through both theoretical development and computational testing as quantum field theory continues to evolve and mature.