Formula for quantum physics
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Formula for Quantum Physics
Introduction to Quantum Theory Formulation
Quantum theory (QT) is traditionally formulated using abstract mathematical constructs such as Hilbert spaces, state vectors, and unitary operators. However, it can also be derived from simpler physical requirements. This approach is akin to deriving the structure of Minkowski space-time in special relativity from the principles of relativity and light speed invariance. By focusing on elementary assumptions about preparations, transformations, and measurements, the full formalism of QT can be derived, providing insights into the physical origins of quantum state spaces and suggesting potential modifications to QT1.
Key Equations in Quantum Physics
Schrödinger, Klein-Gordon, and Dirac Equations
The three main equations in quantum physics—the Schrödinger, Klein-Gordon, and Dirac equations—can be generalized to include nonlinear terms characterized by exponents depending on an index ( q ). These generalized equations maintain the standard linear forms in the limit as ( q ) approaches 1. Remarkably, these equations share a common soliton-like traveling solution expressed through the ( q )-exponential function, which is significant in nonextensive statistical mechanics. Importantly, the Einstein energy-momentum relation is preserved across all values of ( q )2.
Exponential Operators in Quantum Dynamics
In quantum theories, the evolution equations often involve exponential operators with constant or variable exponents, typically in the form of (-iH\delta t), where ( H ) represents Hamiltonians. These exponential operators are fundamental to quantum dynamics, even if they are sometimes obscured by the complexities of perturbative calculations3.
Relativistic Wave Equations
The classical relativistic relationship between energy and momentum leads to wave equations in quantum theory. For instance, the equation ( { pt^2 - px^2 - py^2 - pz^2 - m^2 } \psi = 0 ) transforms into a wave equation where the momentum operators are represented as ( i\hbar \partial/\partial t ) and (-i\hbar \partial/\partial x ). By incorporating particle spin, wave equations consistent with relativistic principles can be derived without involving complex square roots, providing a satisfactory description of particles like electrons and positrons4.
Quantum Fluctuations and Energy
A compact formula has been developed to calculate quantum fluctuations of energy in small subsystems of a hot and relativistic gas. This formula shows increased fluctuations for smaller subsystems, aligning with the energy fluctuations in the canonical ensemble for larger sizes. This expression is useful in various areas of physics, including the study of relativistic heavy-ion collisions and other high-temperature, high-velocity matter5.
Geometric and Algebraic Approaches
Quantum theory can also be formulated using geometric approaches, starting from the set of states. This method provides equations of motion and formulas for probabilities of physical quantities. The geometric approach can be extended to Jordan algebras, generalizing the algebraic approach to quantum theory. This formulation is particularly useful in scattering theory and provides a heuristic proof of decoherence7.
Logical Foundations and Quantum Probability
The phenomena of indeterminism and interference in quantum mechanics can be explained using an extension of classical mathematical logic. In this framework, statements are represented by Hermitian matrices that undergo unitary transformations of truth values. This approach leads to the derivation of the formula for quantum probability and introduces a new method of quantization based on invariants of translations in space8.
Quantum Einstein Equations
The quantum Einstein equations, derived from Bohmian quantum gravity, generalize the classical Einstein equations of gravity. This non-classical geometrodynamics describes the time evolution of a three-geometry and matter fields on a three-manifold, determined by a velocity law defined by the wave function. The wave function satisfies the Wheeler-DeWitt equation, and the resulting dynamics can be cast into the form of the Einstein field equations, with a novel contribution to the energy-momentum tensor from the quantum potential9.
Conclusion
Quantum physics encompasses a range of fundamental equations and principles that describe the behavior of particles and fields at the quantum level. From the Schrödinger, Klein-Gordon, and Dirac equations to the quantum Einstein equations, these formulations provide a comprehensive framework for understanding quantum phenomena. By exploring both traditional and alternative approaches, such as geometric and logical foundations, researchers continue to deepen our understanding of the quantum world.
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A derivation of quantum theory from physical requirements
Quantum theory can be derived from five simple physical requirements, providing insights into the physical origin of quantum state spaces and suggesting possibilities for consistent modifications of quantum theory.
Highly CitedNonlinear relativistic and quantum equations with a common type of solution.
The Schrödinger, Klein-Gordon, and Dirac equations can be generalized with a common solitonlike traveling solution, preserving the Einstein energy-momentum relation for arbitrary values of q.
Highly CitedExponential Formulae in Quantum Theories
Exponential operators with constant or variable exponents (-iHdelta t)(where H are Hamiltonians) are the most typical elements of quantum dynamics, even if sometimes implicit or difficult to perceive behind too many details of perturbative calculations.
Relativistic Wave Equations
This paper presents a new wave equation for particles with spins greater than half a quantum, potentially benefiting future particle discovery and mathematical research.
Highly CitedA compact formula for the quantum fluctuations of energy
Quantum fluctuations of energy in small subsystems of a hot and relativistic gas increase, but agree with canonical ensemble if the size is large enough.
Physical Aspects of Quantum Mechanics
THE purpose of this communication is not to give a report on the present status of quantum mechanics. Such a report has recently been published by W. Heisenberg, the founder of the new theory (Die Naturwissenschaften, 45, 989, 1926). Here we shall make an attempt to understand the physical significance of the quantum theoretical formulae.
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