Hello everyone. This was seen on a Facebook group . Any comments/expertise are welcome.

A Quantum phase space-based approach to unification

The theory of relativity unifies space and time through the concept of spacetime, and changes of reference frames are described by Lorentz transformations, which "mix space and time." In this unification approach, energy-momentum is unified with spacetime through the concept of quantum phase space, and changes of reference frames are described with linear canonical transformations, which "mix energy-momentum and spacetime. "

The concept of phase space originates from the Hamiltonian analytical formulation of classical mechanics. This concept has important applications, for instance, in statistical physics, but it is not compatible with the uncertainty relations of quantum physics. The search for a solution to this problem led to the introduction of the concept of Quantum Phase Space (QPS).

A further challenge in introducing Quantum Phase Space (QPS) involves identifying its associated symmetry group. This group can be shown to be identifiable with the group of Linear Canonical Transformations (LCTs) which is isomorphic to a symplectic group. These LCTs are known in signal processing and optics as integral transforms that generalize fractional Fourier transforms. However, it can be demonstrated that they are equivalent to linear transformations that keep quantum canonical commutation relations covariant. Their relativistic and multidimensional generalizations can then be used to describe changes of reference frames in physics, thereby generalizing De Sitter, Poincaré, and Lorentz transformations, potentially incorporating quantum and gravitational effects.

The illustrative picture shows that the “effects of the mixing between spacetime and energy-momentum"—considered to be potential quantum and gravitational effects—are linked to the extremely small values of the product of the Planck and cosmological constants, and the ratio of the gravitational constant to the speed of light cubed. Consequently, these effects might only be detectable at very large distances or with very high energies." It is also assumed that the approach corresponds to a spacetime whose vacuum is a De Sitter space with a (1,4) signature, which could be in agreement with a positive, non-zero cosmological constant.

The mathematical formulations of physical theories rely on the concepts of space and symmetry groups. We can consider using the concept of QPS and its corresponding symmetry group, formed by LCTs, to better reformulate relativistic quantum physics. This reformulation could aid in the search for solutions to open problems in physics and cosmology, such as: quantum time operator, quantum measurement, unification of interactions, quantum gravity, neutrino oscillations and masses, matter-antimatter asymmetry, dark energy, and dark matter, etc.

source : https://www.facebook.com/photo/?fbid=644632251991561&set=g.1790761467631151

20 minutes ago, Khanzhoren said:

The concept of phase space originates from the Hamiltonian analytical formulation of classical mechanics. This concept has important applications, for instance, in statistical physics, but it is not compatible with the uncertainty relations of quantum physics. The search for a solution to this problem led to the introduction of the concept of Quantum Phase Space (QPS).

I wouldn't say it's not compatible. There is a correspondence through the Poisson bracket. One closely replicates the other, but there is no \( \hbar \) in the classical Poisson brackets, of course.

As to QPS, do you mean the Weyl representation of quantum mechanics with negative probabilities and all that?

Theories that cover different scales of phenomena don't have to be "compatible". One must be the limit of the other one.

Isn't this a speculation?

EDIT: The Weyl-Wigner representation. This:

https://en.wikipedia.org/wiki/Phase-space_formulation

Edited August 14Aug 14 by joigus
minor addition

36 minutes ago, Khanzhoren said:

This was seen on a Facebook group . Any comments/expertise are welcome.

Youtube and facebook but not a journal reference probably means it’s someone doing “vibe physics” with an AI chatbot.

54 minutes ago, Khanzhoren said:

The concept of phase space originates from the Hamiltonian analytical formulation of classical mechanics. This concept has important applications, for instance, in statistical physics, but it is not compatible with the uncertainty relations of quantum physics. The search for a solution to this problem led to the introduction of the concept of Quantum Phase Space (QPS).

Perhaps you wouln't mind outlining that you understand by 'phase space' as the are several definitions available.

At least one is to do withs statistical physics and has 2s dimensions

One other is to do with differential equations and has c+1 dimensions where

where s is the number of degrees of freedom or available and c is the number of derivatives (in calculus) available and either may tend to infinity.

On 8/14/2025 at 3:24 PM, joigus said:

I wouldn't say it's not compatible. There is a correspondence through the Poisson bracket. One closely replicates the other, but there is no ℏ in the classical Poisson brackets, of course.

As to QPS, do you mean the Weyl representation of quantum mechanics with negative probabilities and all that?

Theories that cover different scales of phenomena don't have to be "compatible". One must be the limit of the other one.

Isn't this a speculation?

EDIT: The Weyl-Wigner representation. This:

https://en.wikipedia.org/wiki/Phase-space_formulation

According to the uncertainty relations, one cannot simultaneously have exact values for both momentum and position. Yet, the very definition of a classical phase space pre-supposes the opposite. In my opinion, therefore, the existence of negative probability densities reflects this problem that exists from the very definition of the classical phase space. I acknowledge that this approach, which uses these 'quasiprobabilities,' yields results... but it's probably not the best approach to maintain the classical definition for the phase space itself.

In my opinion, if the relationship of this quantum phase space with the canonical commutation relations and linear canonical transformations is considered directly, one may defines it directly as the set of momenta and coordinates operators that are related to each other by these linear canonical transformations (which leave the canonical commutation relations covariant.)

Edited August 15Aug 15 by Khanzhoren

1 hour ago, Khanzhoren said:

According to the uncertainty relations, one cannot simultaneously have exact values for both momentum and position. Yet, the very definition of a classical phase space pre-supposes the opposite. In my opinion, therefore, the existence of negative probability densities reflects this problem that exists from the very definition of the classical phase space. I acknowledge that this approach, which uses these 'quasiprobabilities,' yields results... but it's probably not the best approach to maintain the classical definition for the phase space itself.

In my opinion, if the relationship of this quantum phase space with the canonical commutation relations and linear canonical transformations is considered directly, one may defines it directly as the set of momenta and coordinates operators that are related to each other by these linear canonical transformations (which leave the canonical commutation relations covariant.)

Why linear? Canonical transformations don't have to be linear. Take, eg,

\[ Q=q+\cos p \]

\[ P=p \]

for a one particle system moving in one direction. The brackets are conserved, and yet, the transformation is non-linear.

Canonical transformations are those that leave the canonical commutation relations invariant, not covariant.

A quantity is not "left" covariant. It is either covariant or it isn't. We don't say it's left covariant because it changes, it is not "left" anywhere. It is taken somewhere else in a particular way we call "covariant". It is called "covariant" because it varies "with the vectors" or "in the same way as vectors".