What is the purpose of this paper?

It is expected that this propagation has many features in common with the propagation of waves in continuous matter and the point of this paper is to demonstrate such a feature, namely a finite bound for the group velocity of a system with finite range interaction.

What is the sinteraction of a particle?

Thus the authors demand that Φ satisfies:1. Φ(X) is Hermitian for X C Έ\\ 2. Φ(X + a) = τa Φ(X) for X C Έv and a e Έv. 3. The union #φ of all X such that X 9 0 and Φ(X) φ 0 is a finite subset of Έv. [Physically only particles situated at the points x e Rφ have a nonzero sinteraction with a particle at the origin.]

What is the stf for the group of space-time automorphisms?

Finally for each finite range interaction Φ, stf is asymptotically abelian for the group of space-time automorphisms (x, t)eΈvxR->τxτf in the cone Cφ= {(x, ί); M > Vφ\\t\\}, i.e.lim \\\\[τxτf(A),B ] \\ \\ = Q |f|-xx>\\x\\>

What is the commutator of a finite system?

Each monomial, M, in Dι(X) is of the formTT yeXHence τfτχ(M)=γ\\τfτx+y(aJ(y)).yeXThe commutator [τf τx(M\\ B~] will have N(X) terms (the number of points in X), each obtained by taking the commutator [τf τΛ+y(αί(y)), B~] and leaving the other elements in τfτx(M) as coefficients.

Open AccessJournal Article10.1007/BF01645779

The Finite Group Velocity of Quantum Spin Systems

Elliott H. Lieb, +1 more

- 01 Sep 1972

- Communications in Mathematical Physics

- Vol. 28, Iss: 3, pp 251-257

1.7K

TL;DR: In this paper, it was shown that information can propagate in a quantum spin system only with a finite group velocity, where μ(ν) > 0, where ρ is the group velocity.

Abstract: It is shown that if Ф is a finite range interaction of a quantum spin system, τ t Ф the associated group of time translations, τ x the group of space translations, and A, B local observables, then $$ \mathop {{\text{lim}}}\limits_{\mathop {\left| t \right| \to \infty }\limits_{\left| x \right| > v\left| t \right|} } \left\| {\left[ {\tau _t^\varphi \tau \left( A \right),B} \right]} \right\|e^{\mu \left( v \right)t} = 0$$ (1) whenever v is sufficiently large (ν > V Ф ,) where μ(ν) > 0. The physical content of the statement is that information can propagate in the system only with a finite group velocity.

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Most frequently asked questions

1. What have the authors contributed in "The finite group velocity of quantum spin systems" ?

In this paper, it was shown that for a large class of translationally invariant interactions, time translations of quantum spin systems can be defined as automorphisms of a C * -algebra, i.e. the abstract algebra generated by the spin operators.

2. what is the commutator for a finite range?

For each finite range interaction Φ there exists a finite group velocity Vφ and a strictly positive increasing function μ such that for v>Vφlim eμ(v^ || [τf τx(A), B]\\\\ =0 | f | ->oo\\*\\>v\\t\\for all strictly local A and B.Proof.

3. What is the commutator for a finite system?

For each X in this sum, the corresponding Φ(X) can be written as a polynomial in the set of elements τy(a^ yeX, j=l,2, ...,N2. The commutator Dt(X) =

4. What is the definition of the abstract algebra?

In [2] it was demonstrated that for a large class of translationally invariant interactions, time translations of quantum spin systems can be defined as automorphisms of a C*-algebra, j/, of quasi-local observables, i.e. the abstract algebra generated by the spin operators.

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- 02 Apr 2014

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References

Journal Article•10.1007/BF01646554

Time development of quantum lattice systems

Mary Beth Ruskai

- 01 Sep 1971

- Communications in Mathematical Physics

TL;DR: In this article, the time development of quantum lattice systems is studied with a weaker assumption on the growth of the potential than has been considered previously, and it is shown that the time complexity of the lattice system can be reduced with respect to the growth rate of potential.

...read moreread less

•Journal Article•10.1007/BF01654286

Statistical mechanics of quantum spin systems. III

Derek W. Robinson

- 01 Jan 1968

- Communications in Mathematical Physics

TL;DR: In this paper, the authors prove that each tangent functional to the graph of P defines a set of translationally invariant thermodynamic expectation values, i.e., an equilibrium state over a suitably chosen algebra.

...read moreread less

Journal Article•10.1007/BF01648328

On certain non-relativistic quantized fields

R. F. Streater

- 01 Jun 1968

- Communications in Mathematical Physics

TL;DR: In this article, the authors considered a polynomial-time interaction Hamiltonian with smeared Fermion field operators (the smearing function providing an ultraviolet cut-off) and showed that time-displacements define a one-parameter group of automorphisms of the field algebra att=0, which acts continuously in the time-parameters.

...read moreread less

The Finite Group Velocity of Quantum Spin Systems

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AI Agents for this Paper

Most frequently asked questions

1. What have the authors contributed in "The finite group velocity of quantum spin systems" ?

2. what is the commutator for a finite range?

3. What is the commutator for a finite system?

4. What is the definition of the abstract algebra?

5. What is the purpose of this paper?

6. What is the sinteraction of a particle?

7. What is the stf for the group of space-time automorphisms?

8. What is the commutator of a finite system?

Citations

Anyons in an exactly solved model and beyond

Colloquium: Area laws for the entanglement entropy

A bound on chaos

Many-Body Localization and Thermalization in Quantum Statistical Mechanics

Many body localization and thermalization in quantum statistical mechanics

References

Time development of quantum lattice systems

Statistical mechanics of quantum spin systems. III

On certain non-relativistic quantized fields

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