How does the Heisenberg uncertainty relation relate to Lie algebras?

Short Answer

The Heisenberg uncertainty relation arises directly from the non-commutativity of position and momentum operators, which form a Lie algebra known as the Heisenberg algebra.

The Canonical Commutation Relation

The fundamental relation in quantum mechanics is:

[\hat{x}, \hat{p}] = \hat{x}\hat{p} - \hat{p}\hat{x} = i\hbar

This is the defining relation of the Heisenberg algebra \mathfrak{h}_1.

The Heisenberg Algebra

The Heisenberg algebra is a 3-dimensional Lie algebra with generators {X, P, I} satisfying:

[X, P] = I, \quad [X, I] = 0, \quad [P, I] = 0

In quantum mechanics:

• X \mapsto \hat{x} (position)
• P \mapsto \hat{p} (momentum)
• I \mapsto i\hbar \cdot \mathbf{1} (central element)

From Commutator to Uncertainty

For any two observables \hat{A} and \hat{B}, the generalized uncertainty relation states:

\Delta A \cdot \Delta B \geq \frac{1}{2}|\langle [\hat{A}, \hat{B}] \rangle|

where \Delta A = \sqrt{\langle \hat{A}^2 \rangle - \langle \hat{A} \rangle^2} is the standard deviation.

Applying to Position and Momentum

Since [\hat{x}, \hat{p}] = i\hbar:

\Delta x \cdot \Delta p \geq \frac{\hbar}{2}

This is the Heisenberg uncertainty principle.

The Lie Algebra Perspective

Why Non-Commutativity Matters

In a Lie algebra, the Lie bracket [A, B] measures how much A and B fail to commute. When [A, B] \neq 0:

1. A and B cannot be simultaneously diagonalized
2. They cannot have simultaneous eigenstates
3. Measuring one disturbs the other

Weyl Algebra

The algebraic structure can also be viewed as the Weyl algebra, the associative algebra generated by x and \partial_x with:

[\partial_x, x] = 1

This is the algebraic foundation of quantum mechanics.

Higher Dimensions

For n degrees of freedom, we have the Heisenberg algebra \mathfrak{h}_n with:

[\hat{x}i, \hat{p}_j] = i\hbar \delta{ij}

This is a (2n+1)-dimensional nilpotent Lie algebra.

Connection to Symplectic Geometry

The classical limit corresponds to Poisson brackets:

{x, p} = 1

This defines a symplectic structure on phase space. Quantization replaces:

{A, B} \to \frac{1}{i\hbar}[\hat{A}, \hat{B}]

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References

• W. Heisenberg, “Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik,” Zeitschrift für Physik 43, 172–198 (1927) — Original uncertainty paper
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 2nd ed. (Addison-Wesley, 2011), Chapter 1
• B. C. Hall, Lie Groups, Lie Algebras, and Representations, 2nd ed. (Springer, 2015), Chapter 3
• V. S. Varadarajan, Geometry of Quantum Theory, 2nd ed. (Springer, 2007)
• A. Kirillov, An Introduction to Lie Groups and Lie Algebras (Cambridge University Press, 2008)
• nLab: Heisenberg Lie algebra