3-Heisenberg-Robertson-Schrodinger Uncertainty Principle

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3-Heisenberg-Robertson-Schrodinger Uncertainty Principle

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K. MAHESH KRISHNA^*

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19 October 2024

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25 October 2024

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Abstract

Let X be a 3-product space. Let A : D(A) ⊆ X → X , B : D(B) ⊆ X → X and C : D(C) ⊆ X → X be possibly unbounded 3-self-adjoint operators. Then for all x ∈ D(ABC) ∩ D(ACB) ∩ D(BAC) ∩ D(BCA) ∩ D(CAB) ∩ D(CBA) with ⟨x, x, x⟩ = 1, we show that (1) ∆x(3, A)∆x(3, B)∆x(3, C) ≥ |⟨(ABC − aBC − bAC − cAB)x, x, x⟩ + 2abc|, where ∆x(3, A) := ∥Ax − ⟨Ax, x, x⟩x∥, a := ⟨Ax, x, x⟩, b := ⟨Bx, x, x⟩, c := ⟨Cx, x, x⟩. We call Inequality (1) as 3-Heisenberg-Robertson-Schrodinger uncertainty principle. Classical HeisenbergRobertson-Schrodinger uncertainty principle (by Schrodinger in 1930) considers two operators whereas Inequality (1) considers three operators.

Keywords:

Subject:

Computer Science and Mathematics - Mathematics

MSC: 46C50; 46B99

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 (also known as variance) 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 (term due to Condon [2]) of Heisenberg derived in 1927 [3]. Recall that, for two linear operators

A : D (A) \subseteq H \to H

and

B : D (B) \subseteq H \to H

, we define

[A, B ≔] A B - B A

and

{A, B} ≔ A B + B A

Theorem 1.

[1,3,4,5,6,7] (Heisenberg-Robertson Uncertainty Principle) Let

A : D (A) \subseteq H \to H

and

B : D (B) \subseteq H \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) [8].

Theorem 2.

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

A : D (A) \subseteq H \to H

and

B : D (B) \subseteq H \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 leads to the following question.

Question 3.

What is the version of Theorem 2 for three operators?

In this note, we answer Question 3 by deriving an uncertainty principle for three operators acting on classes of Banach spaces, using trilinear forms. We note that there are uncertainty principles derived for three operators on Hilbert spaces, but ours differ from them [9,10,11,12,13].

2. 3-Heisenberg-Robertson-Schrodinger Uncertainty Principle

The Heisenberg-Robertson-Schrodinger uncertainty principle requires the inner product to handle two operators; for three operators, we need a 3-product defined as follows.

Definition 1.

Let

X

be a real Banach space with norm

∥ \cdot ∥

. A map

〈 \cdot, \cdot, \cdot 〉 : X \times X \times X \to R

is said to be a 3-product if following conditions hold.

(i): $〈 x, y, z 〉 = 〈 σ (x), σ (y), σ (z) 〉$ for all $x, y, z \in X$ , for all bijections $σ : {x, y, z} \to {x, y, z}$ .
(ii): $〈 α x, y, z 〉 = α 〈 x, y, z 〉$ for all $x, y, z \in X$ , for all $α \in R$ .
(iii): $〈 x + w, y, z 〉 = 〈 x, y, z 〉 + 〈 w, y, z 〉$ for all $x, y, z, w \in X$ .
(iv): $| 〈 x, y, z 〉 | \leq ∥ x ∥ ∥ y ∥ z ∥$ for all $x, y, z \in X$ .

In this case, we say that

X

is a 3-product space.

Following is the standard example we keep in mind.

Example 1.

Let

(Ω, μ)

be a measure space and

L^{3} (Ω, μ)

be the standard real Lebesgue space. Generalized Holder’s inequality says that

\begin{matrix} \int_{Ω} | f_{1} (x) f_{2} (x) f_{3} (x) | d μ (x) & \leq {(\int_{Ω} {| f_{1} (x) |}^{3} d μ (x))}^{\frac{1}{3}} {(\int_{Ω} {| f_{2} (x) |}^{3} d μ (x))}^{\frac{1}{3}} {(\int_{Ω} {| f_{3} (x) |}^{3} d μ (x))}^{\frac{1}{3}} \\ < \infty, \forall f_{1}, f_{2}, f_{3} \in L^{3} (Ω, μ) . \end{matrix}

Therefore

L^{3} (Ω, μ)

is a 3-product space equipped with 3-product

\begin{matrix} 〈 f_{1}, f_{1}, f_{3} 〉 \int_{Ω} f_{1} (x) f_{2} (x) f_{3} (x) d μ (x), \forall f_{1}, f_{2}, f_{3} \in L^{3} (Ω, μ) . \end{matrix}

We next introduce the notion of self-adjointness for operators on 3-product spaces.

Definition 2.

Let

X

be a 3-product space. A possibly unbounded linear operator

A : D (X) \subseteq X \to X

is said to be 3-self-adjoint if

\begin{matrix} 〈 A x, y, z 〉 = 〈 x, A y, z 〉 = 〈 x, y, A z 〉, \forall x, y, z \in D (X) . \end{matrix}

Example 2.

Consider

R^{n}

with the 3-product

\begin{matrix} 〈{(x_{j})}_{j = 1}^{n}, {(y_{j})}_{j = 1}^{n}, {(z_{j})}_{j = 1}^{n}〉 \sum_{j = 1}^{n} x_{j} y_{j} z_{j}, \forall {(x_{j})}_{j = 1}^{n}, {(y_{j})}_{j = 1}^{n}, {(z_{j})}_{j = 1}^{n} \in R^{n} . \end{matrix}

Let

a_{1}, \dots, a_{n}

be any real numbers. Define

\begin{matrix} A : R^{n} ∋ {(x_{j})}_{j = 1}^{n} \mapsto {(a_{j} x_{j})}_{j = 1}^{n} \in R^{n} . \end{matrix}

Then A is 3-self-adjoint.

Example 3.

Consider

ℓ^{3} (N)

(as a real sequence space) with the 3-product

\begin{matrix} 〈{x_{n}}_{n = 1}^{\infty}, {y_{n}}_{n = 1}^{\infty}, {z_{n}}_{n = 1}^{\infty}〉 ≔ \sum_{n = 1}^{\infty} x_{n} y_{n} z_{n}, \forall {x_{n}}_{n = 1}^{\infty}, {y_{n}}_{n = 1}^{\infty}, {z_{n}}_{n = 1}^{\infty} \in ℓ^{3} (N) . \end{matrix}

Let

{x_{n}}_{n = 1}^{\infty}

be a bounded real sequence. Define

\begin{matrix} A : ℓ^{3} (N) ∋ {x_{n}}_{n = 1}^{\infty} \mapsto {a_{n} x_{n}}_{n = 1}^{\infty} \in ℓ^{3} (N) . \end{matrix}

Then A is 3-self-adjoint.

Example 4.

We continue from Example 1. Let

ϕ \in L^{\infty} (Ω, μ)

. Define

\begin{matrix} A : L^{3} (Ω, μ) ∋ f \mapsto A f \in L^{3} (Ω, μ); A f : Ω ∋ α \mapsto (A f) (α) ϕ (α) f (α) \in R . \end{matrix}

Then A is 3-self-adjoint.

Let

A : D (X) \subseteq X \to X

be a 3-self-adjoint operator. For

x \in D (A)

with

〈 x, x, x 〉 = 1

, define the 3-uncertainty of A at the point x as

\begin{matrix} Δ_{x} (3, A) ∥ A x - 〈 A x, x, x 〉 x ∥ . \end{matrix}

Theorem 4.

(3-Heisenberg-Robertson-Schrodinger Uncertainty Principle) Let

X

be a 3-product space. Let

A : D (A) \subseteq X \to X

B : D (B) \subseteq X \to X

and

C : D (C) \subseteq X \to X

be possibly unbounded 3-self-adjoint operators. Then for all

\begin{matrix} x \in D (A B C) \cap D (A C B) \cap D (B A C) \cap D (B C A) \cap D (C A B) \cap D (C B A) \end{matrix}

with

〈 x, x, x 〉 = 1

, we have

\begin{matrix} \frac{1}{27} {(Δ_{x} (3, A) + Δ_{x} (3, B) + Δ_{x} (3, C))}^{3} \geq Δ_{x} (3, A) Δ_{x} (3, B) Δ_{x} (3, C) \geq \\ | 〈 (A B C - 〈 A x, x, x 〉 B C - 〈 B x, x, x 〉 A C - 〈 C x, x, x 〉 A B) x, x, x 〉 + 2 〈 A x, x, x 〉 〈 B x, x, x 〉 〈 C x, x, x 〉 | . \end{matrix}

Proof.

First inequality follows from AM-GM inequality for three positive reals. Given

\begin{matrix} x \in D (A B C) \cap D (A C B) \cap D (B A C) \cap D (B C A) \cap D (C A B) \cap D (C B A) \end{matrix}

with

〈 x, x, x 〉 = 1

, set

\begin{matrix} a ≔ 〈 A x, x, x 〉, b ≔ 〈 B x, x, x 〉, c ≔ 〈 C x, x, x 〉 . \end{matrix}

Then

\begin{matrix} Δ_{x} (3, A) Δ_{x} (3, B) Δ_{x} (3, C) \geq | 〈 A x - a x, B x - b x, C x - c x 〉 | \\ = | 〈 A B C x, x, x 〉 - 〈 (a B C + b A C + c A B) x, x, x 〉 + 〈 (a b C + b c A + c a B) x, x, x 〉 - a b c | \\ = | 〈 A B C x, x, x 〉 - 〈 (a B C + b A C + c A B) x, x, x 〉 + 2 a b c | \\ = | 〈 (A B C - 〈 A x, x, x 〉 B C - 〈 B x, x, x 〉 A C - 〈 C x, x, x 〉 A B) x, x, x 〉 + 2 〈 A x, x, x 〉 〈 B x, x, x 〉 〈 C x, x, x 〉 | . \end{matrix}

□

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3-Heisenberg-Robertson-Schrodinger Uncertainty Principle

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Keywords:

Subject:

1. Introduction

2. 3-Heisenberg-Robertson-Schrodinger Uncertainty Principle

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