What is a non-local Hamiltonian?

[lindberg]

TL;DR

Can someone explain in simple terms (high-school) how a Hamiltonian can be non-local?

If I understand it correctly, the Hamiltonian represents the total energy of the system. Can it be non-local? If yes, doesn't this contradict relativistic locality?

 

[haushofer]

It's nonlocal if the energy (density) at spacetime point x also depends instantaneously on point x+y. Yes, that violates relativity.

This happens e.g. if the hamiltonian contains a term f(x)f(x+y), where f is the general field.

More concrete example: the energy density of an electromagnetic field contains a term E^2(x) with E the electric field. Replace it with E(x)E(x+y) and your energy density (Hamiltonian) becomes nonlocal.

 

[lindberg]

Thank you, @haushofer. I have found it in this paper: https://arxiv.org/abs/1211.5878 (page 4 and 7) and it seemed odd to me.

 

[vanhees71]

Well, look where it had been published...

 

[dextercioby]

Well, look where it had been published...

What do you mean? Is it the same "Oh, 《Foundations of Physics》is really badly peer-reviewed!", or what?

 

[Demystifier]

What do you mean? Is it the same "Oh, 《Foundations of Physics》is really badly peer-reviewed!", or what?

I think @vanhees71 has a low opinion on this journal, probably because it deals with questions which are partially philosophical. I would like to stress that this journal has very distinguished editors. Now it is Carlo Rovelli (most famous for his work in loop quantum gravity), and before him it was the Nobel laureate G. 't Hooft.

 

[vanhees71]

Well, I often wonder, whether this is still a physics journal ;-)).

 

[Demystifier]

Well, I often wonder, whether this is still a physics journal ;-)).

Of course it isn't, it is a foundations of physics journal. Foundations of physics is not physics. More generally, foundations of X studies what is X based on, and no X is based on itself.

 

[lindberg]

@Demystifier so wait, what is Physics based on then?
And back to my original question: isn't it a problem that to formulate an alternative to GR, they give up the locality of the Hamiltonian? It seems like a big problem. Isn't experiment ruling this out?

 

[malawi_glenn]

what is Physics based on then?

Physics is based on the scientific method. An interplay between experiments, measurments, observations with mathematical models and theories.

 

[Demystifier]

@Demystifier so wait, what is Physics based on then?

The answer depends on what kind of answer do you expect. For example, if the question was "What is chemistry based on?", what would be your answer? Would it be - physics?

And back to my original question: isn't it a problem that to formulate an alternative to GR, they give up the locality of the Hamiltonian? It seems like a big problem. Isn't experiment ruling this out?

What experiment? Sure, many experiments rule out non-local Hamiltonians in various contexts, but I am not aware of any experiment that rules it out in the cosmological context.

 

[Demystifier]

Physics is based on the scientific method. An interplay between experiments, measurments, observations with mathematical models and theories.

Yes, but you can hardly find this definition of the scientific method in science books. It's much more likely that you will find it in books on philosophy of science.

 

[malawi_glenn]

Yes, but you can hardly find this definition of the scientific method in science books. It's much more likely that you will find it in books on philosophy of science.

Exactly, the scientific method is not scientific so to say :)

 

[samalkhaiat]

Well, I often wonder, whether this is still a physics journal ;-)).

I remember, in the 1990’s, and at the tea room of the physics department of Birkbeck College, that journal was the subject of almost monthly jokes. The one that I still remember is: “It is good to know that physics is not based on “FOUNDATIONS OF PHYSICS”” Prof. E. Leader

 

[Demystifier]

I remember, in the 1990’s, and at the tea room of the physics department of Birkbeck College, that journal was the subject of almost monthly jokes. The one that I still remember is: “It is good to know that physics is not based on “FOUNDATIONS OF PHYSICS”” Prof. E. Leader

I like the one by Feynman, not about the specific journal but about the whole field. Physics needs philosophy of physics as much as birds need ornithology.

BTW, there are also jokes about mathematical physicists, but I will save them for another occasion.

 

[HomogenousCow]

It's nonlocal if the energy (density) at spacetime point x also depends instantaneously on point x+y. Yes, that violates relativity.

Is this true? I thought one could write down non-local field theories that were still lorentz invariant.

 

[vanhees71]

Can one? Do you have an example for such a theory? Is it consistent with observations?

 

[Demystifier]

Actions containing $1/\Box$ are nonlocal and Lorentz invariant, many effective theories (for otherwise local theories) have this form.

 

[PeterDonis]

Actions containing $1/\Box$ are nonlocal and Lorentz invariant

I get Lorentz invariant--$\Box$ is Lorentz invariant and so therefore is $1 / \Box$--but how is it nonlocal?

 

[vanhees71]

It's not unique, but one can define $1/\Box$ as, e.g., the retarded Green's function. Then it's an operator on an field $J$ defined by
$$\frac{1}{\Box} J(x)=\int_{\mathbb{R}^4} \mathrm{d}^4 x' G_{\text{ret}}(x-x') J(x')$$
with
$$G_{\text{ret}}(x)=\frac{\Theta(x^0)}{2 \pi} \delta(x \cdot x),$$
which is a scalar under proper orthochronous Lorentz transformations. It's nonlocal, because it depends on values of $J$ along the entire past light cone of the event $x$.

 

[Demystifier]

I get Lorentz invariant--$\Box$ is Lorentz invariant and so therefore is $1 / \Box$--but how is it nonlocal?

@vanhees71 explained above.