We propose a unified form of possible generalizations of Heisenberg uncertainty principle for strong interactions, which is based on the product of the envelope of the dynamical equations representing wave packets under conditions of mutual overlapping and nonlocal non-Hamiltonian forces. For conventional associative products of operators, we recover the conventional Heisenberg relations for particles under long range electromagnetic interactions and no appreciable wave overlapping. For nonassociative Lie-admissible products of operators we have (a) a number of generalizations of Heisenberg's equations of increasing algebraic complexity; (b) the capability of representing a correspondingly progressive increase of conditions of wave overlapping and nonlocal, non-Hamiltonian forces; and (c) a corresponding, rather inevitable sequence of generalizations of Heisenberg uncertainty relations. All these possibilities are expressed in a unified way by postulating generalized uncertainty relations whereby the product of operators is that of the envelope of the dynamical equations. For conditions of minimal wave overlapping under strong interactions, such as in the nuclear structure, we expect a very small nonlocal non-Hamiltonian component of the force and a minimal departure from Heisenberg principle. For progressive increase of wave overlappings, such as in the structure of hadrons, or in the structure of stars undergoing gravitational collapse, we expectmore » a corresponding progressive replacement of Heisenberg principle with more general principles. The problem of the experimental resolution of the exact or only approximate validity of Heisenberg principle under strong interactions is considered for a more detailed treatment in a separate paper.« less
We propose a unified form of possible generalizations of Heisenberg uncertainty principle for strong interactions, which is based on the product of the envelope of the dynamical equations representing wave packets under conditions of mutual overlapping and nonlocal non-Hamiltonian forces. For conventional associative products of operators, we recover the conventional Heisenberg relations for particles under long range electromagnetic interactions and no appreciable wave overlapping. For nonassociative Lie-admissible products of operators we have (a) a number of generalizations of Heisenberg's equations of increasing algebraic complexity; (b) the capability of representing a correspondingly progressive increase of conditions of wave overlapping and nonlocal, non-Hamiltonian forces; and (c) a corresponding, rather inevitable sequence of generalizations of Heisenberg uncertainty relations. All these possibilities are expressed in a unified way by postulating generalized uncertainty relations whereby the product of operators is that of the envelope of the dynamical equations. For conditions of minimal wave overlapping under strong interactions, such as in the nuclear structure, we expect a very small nonlocal non-Hamiltonian component of the force and a minimal departure from Heisenberg principle. For progressive increase of wave overlappings, such as in the structure of hadrons, or in the structure of stars undergoing gravitational collapse, we expectmore » a corresponding progressive replacement of Heisenberg principle with more general principles. The problem of the experimental resolution of the exact or only approximate validity of Heisenberg principle under strong interactions is considered for a more detailed treatment in a separate paper.« less
Due to the requirements of the principle of causality in the theory of relativity, one cannot make a device for the simultaneous measuring of the canonical conjugate variables in the conjugate Fourier spaces. Instead of admitting that a particle’s position and its conjugate momentum cannot be accurately measured at the same time, we consider the only probabilities which can be determined when working at subatomic level to be valid. On the other hand, based on Schwinger's action principle and using the quadridimensional form of the unitary transformation generator function of the quantum operators in the paper, the general form of the evolution equation for these operators is established. In the nonrelativistic case one obtains the Heisenberg's type evolution equations which can be particularized to derive Heisenberg's uncertainty relations. The analysis of the uncertainty relations as implicit evolution equations allows us to put into evidence the intrinsic nature of the correlation expressed by these equations in straight relations with the measuring process. The independence of the quantisation postulate from the causal evolution postulate of quantum mechanics is also put into discussion.
This work develops a unified structural explanation of Heisenberg’s uncertainty principle based onthe infinite-dimensional nature of physical reality and its mathematical representation. Quantumstates are intrinsically wave-like and must be described by functions rather than finite-dimensionalvectors, forcing the use of an infinite-dimensional Hilbert space. Within this framework, fourindependent but mutually reinforcing structures emerge: analytic non-commutativity ofmultiplication and differentiation, Fourier duality and the mutual broadening of a function and itstransform, the symplectic geometry of phase space and its minimal area constraint, and thealgebraic rigidity encoded in the Weyl relations and the Stone–von Neumann theorem. Theselayers form a logical chain from finite-dimensional impossibility to infinite-dimensional necessity,revealing the uncertainty principle as a structural feature of the physical world. The paper furtherdiscusses extensions to the generalized uncertainty principle and the infinite-dimensionalstructure of quantum field theory.
The historical development of uncertainty relation, begining with first Heisenberg's thoughts of uncertainty principle is summed up in this thesis. After proving validity of Schwarz inequality general uncertainty relation for two hermitian operators is obtained, and from this general version the validity of Heisenberg uncertainty relation is than proved. The most important part of this work is the obtention of two new uncertainty relations, which are stronger then Heisenberg or Robertson-Schrödinger uncertainty relation, and their specific form for two examples - a free particle in a state discribed by the gaussian wave packet and the linear harmonic oscillator with a wave function in the shape of gaussian packet.
In this short note we show how the Generalised Uncertainty Principle (GUP) and the Extended Uncertainty Principle (EUP), two of the most common generalised uncertainty relations proposed in the quantum gravity literature, can be derived within the context of canonical quantum theory, without the need for modified commutation relations. A generalised uncertainty principle-type relation naturally emerges when the standard position operator is replaced by an appropriate Positive Operator Valued Measure (POVM), representing a finite-accuracy measurement that localises the quantum wave packet to within a spatial regionσg> 0. This length scale is the standard deviation of the envelope function,g, that defines the positive operator valued measure elements. Similarly, an extended uncertainty principle-type relation emerges when the standard momentum operator is replaced by a positive operator valued measure that localises the wave packet to within a region σ̃g>0 in momentum space. The usual generalised uncertainty principle and extended uncertainty principle are recovered by setting σg≃ℏG/c3 , the Planck length, and σ̃g≃ℏΛ/3 , where Λ is the cosmological constant. Crucially, the canonical Hamiltonian and commutation relations, and, hence, the canonical Schrödinger and Heisenberg equations, remain unchanged. This demonstrates that generalised uncertainty principle and extended uncertainty principle phenomenology can be obtained without modified commutators, which are known to lead to various pathologies, including violation of the equivalence principle, violation of Lorentz invariance in the relativistic limit, the reference frame-dependence of the “minimum” length, and the so-called soccer ball problem for multi-particle states.
Abstract Inspired by the generalized uncertainty principle, which adds gravitational effects to the standard description of quantum uncertainty, we extend the exact uncertainty principle approach by Hall and Reginatto (2002 J. Phys. A: Math. Gen. 35 3289 ), and obtain a (quasi)nonlinear Schrödinger equation. This quantum evolution equation of unusual form, enjoys several desired properties like separation of non-interacting subsystems or plane-wave solutions for free particles. Starting with the harmonic oscillator example, we show that every solution of this equation respects the gravitationally induced minimal position uncertainty proportional to the Planck length. Quite surprisingly, our result successfully merges the core of classical physics with non-relativistic quantum mechanics in its extremal form. We predict that the commonly accepted phenomenon, namely a modification of a free-particle dispersion relation due to quantum gravity might not occur in reality.
Within an algebraic framework of non-associative quantum mechanics, a Hilbert space and the corresponding Schroedinger equation have been introduced into the theory as follows. The underlying non-associative operator algebra A is assumed to be flexible and Lie-admissible so that it is compatible with the Heisenberg equation of motion and quantization. A Hilbert space can be introduced into the theory as a faithful representation of the associated Lie algebra A/sup -/ of the Lie-admissible algebra A. However, if A/sup -/ is the standard Heisenberg algebra, and if the representation is irreducible, then it is shown that A must be associative, reproducing the standard associative quantum mechanics. Some discussions to circumvent this No-Go theorem are discussed. Especiallyyu if we are interested in the formulation of infinite component wave equations, then this difficulty can be avoided.