# Bounded perturbations of the Heisenberg commutation relation via dilation theory

@inproceedings{Gerhold2022BoundedPO, title={Bounded perturbations of the Heisenberg commutation relation via dilation theory}, author={M. Gerhold and Orr Shalit}, year={2022} }

. We extend the notion of dilation distance to strongly continuous one-parameter unitary groups. If the dilation distance between two such groups is ﬁnite, then these groups can be represented on the same space in such a way that their generators have the same domain and are in fact a bounded perturbation of one another. This result extends to d -tuples of one-parameter unitary groups. We apply our results to the Weyl canonical commutation relations, and as a special case we recover the result…

## References

SHOWING 1-10 OF 11 REFERENCES

### Continuous perturbations of noncommutative Euclidean spaces and tori

- Mathematics
- 2016

We prove a noncompact version of Haagerup and R{\o}rdam's result about continuous paths of the rotation $C^*$-algebras. It gives a continuous Moyal deformation of Euclidean plane. Moveover, the…

### Dilations of unitary tuples

- Mathematics, Computer ScienceJournal of the London Mathematical Society
- 2021

The space of all d ‐tuples of unitaries u=(u1,…,ud) is studied using dilation theory and matrix ranges to find the minimal dilation constant c=c(u,v) such that u≺cv, and the metrics dD and dHR are equivalent to the Hausdorff distance between the matrix ranges of the tuples.

### On the Stone — von Neumann Uniqueness Theorem and Its Ramifications

- Physics
- 2001

In the mid to late 1920s, the emerging theory of quantum mechanics had two main competing (and, initially, mutually antagonistic) formalisms — the wave mechanics of E. Schrodinger [61] and the matrix…

### Quantum Theory for Mathematicians

- Mathematics
- 2013

1 The Experimental Origins of Quantum Mechanics.- 2 A First Approach to Classical Mechanics.- 3 A First Approach to Quantum Mechanics.- 4 The Free Schrodinger Equation.- 5 A Particle in a Square…

### An Introduction to Quantum Stochastic Calculus

- Mathematics
- 1992

"Elegantly written, with obvious appreciation for fine points of higher mathematics...most notable is [the] author's effort to weave classical probability theory into [a] quantum framework." - The…

### An Invitation To C*-Algebras

- Mathematics
- 1976

1 Fundamentals.- 1.1. Operators and C*-algebras.- 1.2. Two density theorems.- 1.3. Ideals, quotients, and representations.- 1.4. C*-algebras of compact operators.- 1.5. CCR and GCR algebras.- 1.6.…

### Dilations of $q$-Commuting Unitaries

- Mathematics
- 2019

Let $q = e^{i \theta} \in \mathbb{T}$ (where $\theta \in \mathbb{R}$), and let $u,v$ be $q$-commuting unitaries, i.e., $u$ and $v$ are unitaries such that $vu = quv$. In this paper we find the…

### Perturbations of the rotation $C^{\ast}$-algebras and of the Heisenberg commutation relation

- Mathematics
- 1995