Nonlinear Heisenberg-Robertson-Schrodinger Uncertainty Principle

Preprint

Article

Nonlinear Heisenberg-Robertson-Schrodinger Uncertainty Principle

Altmetrics

Downloads

135

Views

144

Comments

K. MAHESH KRISHNA^*

This version is not peer-reviewed

Submitted:

05 February 2024

Posted:

06 February 2024

You are already at the latest version

Alerts

Abstract

We derive an uncertainty principle for Lipschitz maps acting on subsets of Banach spaces. We show that this nonlinear uncertainty principle reduces to the Heisenberg-Robertson-Schrodinger uncertainty principle for linear operators acting on Hilbert spaces.

Keywords:

Subject: Computer Science and Mathematics - Analysis

1. Introduction

Let

H

be a complex Hilbert space and A be a possibly unbounded self-adjoint linear operator defined on domain

D (A) \subseteq H

. For

h \in D (A)

with

∥ h ∥ = 1

, define the uncertainty of A at the point h as

\begin{matrix} Δ_{h} (A) : = ∥ A h - 〈 A h, h 〉 h ∥ = \sqrt{{∥ A h ∥}^{2} - {〈 A h, h 〉}^{2}} . \end{matrix}

In 1929, Robertson [1] derived the following mathematical form of the uncertainty principle (also known as uncertainty relation) of Heisenberg derived in 1927 [2] (English translation of 1927 original article by Heisenberg). Recall that for two linear operators

A : D (A) \to H

and

B : D (B) \to H

, we define

[A, B] : = A B - B A

and

{A, B} : = A B + B A

Theorem 1.

[1,2,3,4,5,6] (Heisenberg-Robertson Uncertainty Principle) Let

A : D (A) \to H

and

B : D (B) \to H

be self-adjoint operators. Then for all

h \in D (A B) \cap D (B A)

with

∥ h ∥ = 1

, we have

\begin{matrix} \frac{1}{2} (Δ_{h} {(A)}^{2} + Δ_{h} {(B)}^{2}) \geq \frac{1}{4} {(Δ_{h} (A) + Δ_{h} (B))}^{2} \geq Δ_{h} (A) Δ_{h} (B) \geq \frac{1}{2} | 〈 [A, B] h, h 〉 | . \end{matrix}

(1)

In 1930, Schrodinger improved Inequality (1) [7] (English translation of 1930 original article by Schrodinger).

Theorem 2.

[7] (Heisenberg-Robertson-Schrodinger Uncertainty Principle) Let

A : D (A) \to H

and

B : D (B) \to H

be self-adjoint operators. Then for all

h \in D (A B) \cap D (B A)

with

∥ h ∥ = 1

, we have

\begin{matrix} Δ_{h} (A) Δ_{h} (B) \geq | 〈 A h, B h 〉 - 〈 A h, h 〉 〈 B h, h 〉 | = \frac{\sqrt{{| 〈 [A, B] h, h 〉 |}^{2} + {| 〈 {A, B} h, h 〉 - 2 〈 A h, h 〉 〈 B h, h 〉 |}^{2}}}{2} . \end{matrix}

Theorem 2 promotes the following question.

Question 1.

What is the nonlinear (even Banach space) version of Theorem 2?

In this short note, we answer Question 1 by deriving an uncertainty principle for Lipschitz maps acting on subsets of Banach spaces. We note that there is a Banach space version of uncertainty principle by Goh and Goodman [8] which differs from the results in this paper. We also note that nonlinear Maccone-Pati uncertainty principle is derived in [9].

2. Nonlinear Heisenberg-Robertson-Schrodinger Uncertainty Principle

Let

X

be a Banach space. Recall that the collection of all Lipschitz functions

f : X \to C

satisfying

f (0) = 0

, denoted by

X^{#}

is a Banach space [10] w.r.t. the Lipschitz norm

\begin{matrix} {∥ f ∥}_{{Lip}_{0}} : = sup_{x, y \in X, x \neq y} \frac{| f (x) - f (y) |}{∥ x - y ∥} . \end{matrix}

Let

M \subseteq X

be a subset such that

0 \in M

. Let

A : M \to X

be a Lipschitz map such that

A (0) = 0

. Given

x \in M

and

f \in X^{#}

satisfying

f (x) = 1

, we define two uncertainties of A at

(x, f) \in M \times X^{#}

\begin{matrix} Δ (A, x, f) : = ∥ A x - f (A x) x ∥, \\ {\nabla (f, A, x) : = ∥ f A - f (A x) f ∥}_{{Lip}_{0}} . \end{matrix}

Theorem 3.

(Nonlinear Heisenberg-Robertson-Schrodinger Uncertainty Principle) Let

X

be a Banach space and

M, N \subseteq X

be subsets such that

0 \in M \cap N

. Let

A : M \to X

B : N \to X

be Lipschitz maps such that

A (0) = B (0) = 0

. Then for all

x \in M \cap N

and

f \in X^{#}

satisfying

f (x) = 1

, we have

\begin{matrix} \frac{1}{2} (\nabla {(f, A, x)}^{2} + Δ {(B, x, f)}^{2}) & \geq \frac{1}{4} {(\nabla (f, A, x) + Δ (B, x, f))}^{2} \\ \geq \nabla (f, A, x) Δ (B, x, f) \geq | f (A B x) - f (A x) f (B x) | . \end{matrix}

Proof.

\begin{matrix} \nabla (f, A, x) Δ (B, x, f) & = {∥ f A - f (A x) f ∥}_{{Lip}_{0}} ∥ B x - f (B x) x ∥ \\ \geq | [f A - f (A x) f] [B x - f (B x) x] | \\ = | f (A B x) - f (B x) f (A x) - f (A x) f (B x) + f (A x) f (B x) f (x) | \\ = | f (A B x) - f (B x) f (A x) - f (A x) f (B x) + f (A x) f (B x) \cdot 1 | \\ = | f (A B x) - f (A x) f (B x) | . \end{matrix}

□

Corollary 1.

Let

X

be a Banach space and

M, N \subseteq X

be subsets such that

0 \in M \cap N

. Let

A : M \to X

B : N \to X

be Lipschitz maps such that

A (0) = B (0) = 0

. Then for all

x \in M \cap N

and

f \in X^{#}

satisfying

f (x) = 1

, we have

\begin{matrix} \nabla (f, A, x) Δ (B, x, f) + {∥ f B ∥}_{{Lip}_{0}} Δ (A, x, f) \geq | f ([A, B] x) | . \end{matrix}

Proof.

Note that

\begin{matrix} \nabla (f, A, x) Δ (B, x, f) & \geq | f (A B x) - f (A x) f (B x) | \\ = | f (A B x - B A x) + f (B A x) - f (A x) f (B x) | \\ = | f ([A, B] x) + (f B) [A x - f (A x) x] | \\ \geq | f ([A, B] x) | - | (f B) [A x - f (A x) x] | \\ \geq | f ([A, B] x) | - {∥ f B ∥}_{{Lip}_{0}} ∥ A x - f (A x) x ∥ \\ = | f ([A, B] x) | - {∥ f B ∥}_{{Lip}_{0}} Δ (A, x, f) . \end{matrix}

□

Corollary 2.

Let

X

be a Banach space and

M, N \subseteq X

be subsets such that

0 \in M \cap N

. Let

A : M \to X

B : N \to X

be Lipschitz maps such that

A (0) = B (0) = 0

. Then for all

x \in M \cap N

and

f \in X^{#}

satisfying

f (x) = 1

, we have

\begin{matrix} \nabla (f, A, x) Δ (B, x, f) + {∥ f B ∥}_{{Lip}_{0}} Δ (A, x, - f) \geq | f ({A, B} x) | . \end{matrix}

Proof.

Note that

\begin{matrix} \nabla (f, A, x) Δ (B, x, f) & \geq | f (A B x) - f (A x) f (B x) | \\ = | f (A B x + B A x) - f (B A x) - f (A x) f (B x) | \\ = | f ({A, B} x) - (f B) [A x + f (A x) x] | \\ \geq | f ({A, B} x) | - | (f B) [A x + f (A x) x] | \\ \geq | f ({A, B} x) | - {∥ f B ∥}_{{Lip}_{0}} ∥ A x - (- f) (A x) x ∥ \\ = | f ({A, B} x) | - {∥ f B ∥}_{{Lip}_{0}} Δ (A, x, - f) . \end{matrix}

□

Corollary 3.

(Functional Heisenberg-Robertson-Schrodinger Uncertainty Principle) Let

X

be a Banach space with dual

X^{*}

A : D (A) \to X

and

B : D (B) \to X

be linear operators. Then for all

x \in D (A B) \cap D (B A)

and

f \in X^{*}

satisfying

f (x) = 1

, we have

\begin{matrix} \frac{1}{2} (\nabla {(f, A, x)}^{2} + Δ {(B, x, f)}^{2}) & \geq \frac{1}{4} {(\nabla (f, A, x) + Δ (B, x, f))}^{2} \\ \geq \nabla (f, A, x) Δ (B, x, f) \geq | f (A B x) - f (A x) f (B x) | . \end{matrix}

Corollary 4.

Let

X

be a Banach space,

A : D (A) \to X

and

B : D (B) \to X

be linear operators. Then for all

x \in D (A B) \cap D (B A)

and

f \in X^{*}

satisfying

f (x) = 1

, we have

\begin{matrix} \nabla (f, A, x) Δ (B, x, f) + ∥ f B ∥ Δ (A, x, f) \geq | f ([A, B] x) | . \end{matrix}

Corollary 5.

Let

X

be a Banach space,

A : D (A) \to X

and

B : D (B) \to X

be linear operators. Then for all

x \in D (A B) \cap D (B A)

and

f \in X^{*}

satisfying

f (x) = 1

, we have

\begin{matrix} \nabla (f, A, x) Δ (B, x, f) + ∥ f B ∥ Δ (A, - x, f) = \nabla (f, A, x) Δ (B, x, f) + ∥ f B ∥ Δ (A, x, - f) \geq | f ({A, B} x) | . \end{matrix}

Corollary 6.

Theorem 2 follows from Theorem 3.

Proof.

Let

H

be a complex Hilbert space.

A : D (A) \to H

and

B : D (B) \to H

be self-adjoint operators. Let

h \in D (A B) \cap D (B A)

with

∥ h ∥ = 1

. Define

X : = H

M : = D (A)

N : = D (B)

x : = h

and

\begin{matrix} f : H ∋ u \mapsto f (u) : = 〈 u, h 〉 \in C . \end{matrix}

Then

\begin{matrix} Δ (B, x, f) = Δ (B, h, f) = ∥ B h - f (B h) h ∥ = ∥ B h - 〈 B h, h 〉 h ∥ = Δ_{h} (B), \end{matrix}

\begin{matrix} \nabla (f, A, x) & = \nabla (f, A, h) = ∥ f A - f (A h) f ∥ = ∥ f A - 〈 A h, h 〉 f ∥ \\ = sup_{u \in H, ∥ u ∥ \leq 1} | f (A u) - 〈 A h, h 〉 f (u) | = sup_{u \in H, ∥ u ∥ \leq 1} | 〈 A u, h 〉 - 〈 A h, h 〉 〈 u, h 〉 | \\ = sup_{u \in H, ∥ u ∥ \leq 1} | 〈 u, A h 〉 - 〈 A h, h 〉 〈 u, h 〉 | = sup_{u \in H, ∥ u ∥ \leq 1} | 〈 u, A h - 〈 A h, h 〉 h 〉 | \\ = ∥ A h - 〈 A h, h 〉 h ∥ = Δ_{h} (A) \end{matrix}

and

\begin{matrix} | f (A B x) - f (A x) f (B x) | & = | f (A B h) - f (A h) f (B h) | = | 〈 A B h, h 〉 - 〈 A h, h 〉 〈 B h, h 〉 | \\ = | 〈 B h, A h 〉 - 〈 A h, h 〉 〈 B h, h 〉 | = | 〈 A h, B h 〉 - 〈 A h, h 〉 〈 B h, h 〉 | . \end{matrix}

□

References

Robertson, H.P. The Uncertainty Principle. Phys. Rev. 1929, 34, 163–164. [Google Scholar] [CrossRef]
Heisenberg, W. The physical content of quantum kinematics and mechanics. In Quantum Theory and Measurement; Wheeler, J.A., Zurek, W.H., Eds.; Princeton Series in Physics; Princeton University Press: Princeton, NJ, 1983; pp. 62–84. [Google Scholar]
Cassidy, D.C. Heisenberg, uncertainty and the quantum revolution. Sci. Amer. 1992, 266, 106–112. [Google Scholar] [CrossRef]
von Neumann, J. Mathematical foundations of quantum mechanics; Princeton University Press, Princeton, NJ, 2018; pp. xviii+304. [CrossRef]
Debnath, L.; Mikusiński, P. Introduction to Hilbert spaces with applications; Academic Press, Inc., San Diego, CA, 1999; pp. xx+551.
Ozawa, M. Heisenberg’s original derivation of the uncertainty principle and its universally valid reformulations. Current Science 2015, 109, 2006–2016. [Google Scholar] [CrossRef]
Schrödinger, E. About Heisenberg uncertainty relation (original annotation by A. Angelow and M.-C. Batoni). Bulgar. J. Phys. 1999, 26, 193–203 (2000). Translation of Proc. Prussian Acad. Sci. Phys. Math. Sect. 19 (1930), 296–303. [CrossRef]
Goh, S.S.; Goodman, T.N.T. Uncertainty principles in Banach spaces and signal recovery. J. Approx. Theory 2006, 143, 26–35. [Google Scholar] [CrossRef]
Krishna, K.M. Nonlinear Maccone-Pati Uncertainty Principle. 31 January 2024. [CrossRef]
Weaver, N. Lipschitz algebras; World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018; pp. xiv+458. [CrossRef]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.

MDPI Initiatives

Important Links

Choose an area of interest and we will send you notifications of new preprints at your preferred frequency.

Disclaimer

Nonlinear Heisenberg-Robertson-Schrodinger Uncertainty Principle

Abstract

1. Introduction

2. Nonlinear Heisenberg-Robertson-Schrodinger Uncertainty Principle

References

MDPI Initiatives

Important Links

Subscribe