Understanding Local and Nonlocal Operators in Quantum Field Theory

[samalkhaiat]

@samalkhaiat what sort of "incurable diseases" does Nonlocal theories posses?

Causality and the absence of well-posed Cauchy Problems.

 

[vanhees71]

What is the issue that needs to be settled? The summary is good, it gives the definitions i wanted to see.

This is what I was asking you all the time. I'm glad that the apparent problems are settled now.

 

[MathematicalPhysicist]

Causality and the absence of well-posed Cauchy Problems.

"Well-posed Cauchy problems", does it mean we have more than one solution to the PDE?

 

[martinbn]

"Well-posed Cauchy problems", does it mean we have more than one solution to the PDE?

Or none, or the solutions don't depend continuously on the data. These three are needed for a well-posed problem.

 

[samalkhaiat]

"Well-posed Cauchy problems", does it mean we have more than one solution to the PDE?

Yes and No (as mentioned by martinbn above). Even though in most of the models the resulting equations have no solutions, but you asked about nonlocal theories (the plural of theory) hence the plural of Cauchy problem.

 

[martinbn]

Yes and No (as mentioned by martinbn above). Even though in most of the models the resulting equations have no solutions, but you asked about nonlocal theories (the plural of theory) hence the plural of Cauchy problem.

Side question, why is an "ill-posed Cauchy problem" a problem? If the equations are say elliptic, then a different boundary value problem would be the natural one. And to me elliptic equations are non-local.

 

[Demystifier]

And to me elliptic equations are non-local.

Why?

 

[martinbn]

Why?

A few reasons, for example you cannot localize solutions. You cannot change the solution in some region without changing it everywhere.

 

[samalkhaiat]

Side question, why is an "ill-posed Cauchy problem" a problem? If the equations are say elliptic, then a different boundary value problem would be the natural one. And to me elliptic equations are non-local.

Mathematically, there are no problems with elliptic (hypo-elliptic) differential operator, because (by definition) it possesses a (real) analytic (respectively C^{\infty}) fundamental solution. Laplace and Cauchy-Riemann operators are elliptic; the heat operator is hypo-elliptic.
The operators that govern the dynamical evolution on space-time need to be hyperbolic relative to the future light-cone.

 

[martinbn]

Mathematically, there are no problems with elliptic (hypo-elliptic) differential operator, because (by definition) it possesses a (real) analytic (respectively C^{\infty}) fundamental solution. Laplace and Cauchy-Riemann operators are elliptic; the heat operator is hypo-elliptic.
The operators that govern the dynamical evolution on space-time need to be hyperbolic relative to the future light-cone.

Ok, so you meant that the equations will not be hyperbolic, rather than ill posed Cauchy problem. The heat equation has a well posed Cauchy problem, but I wouldn't call it local.

 

[vanhees71]

That's why in relativistic hydro the Navier Stokes viscous hydro (1st order gradients) is acausal and instable, you have to go to 2nd order gradients ("Israel Stewart").

 

[MathematicalPhysicist]

That's why in relativistic hydro the Navier Stokes viscous hydro (1st order gradients) is acausal and instable, you have to go to 2nd order gradients ("Israel Stewart").

What is "acausal"? the future affects the present? surely you didn't mean "static", right?

 

[vanhees71]

You get faster-than light propagation of sound waves, which is clearly acausal. For some details, see, e.g.,

https://arxiv.org/abs/0807.3120

 

[MathematicalPhysicist]

You get faster-than light propagation of sound waves, which is clearly acausal. For some details, see, e.g.,

https://arxiv.org/abs/0807.3120

Back into the past travel...

 

[samalkhaiat]

Ok, so you meant that the equations will not be hyperbolic, rather than ill posed Cauchy problem.

My statement in #151 meant the absence of causal evolution. Causal evolution means “filling up” the future light-cone with non-intersecting space-like hyper-surfaces, i.e., Cauchy slices. The equations, derived from local relativistic Lagrangian, smoothly and happily take you from one Cauchy slices to the next (see exercise (1) in # 144).

The heat equation has a well posed Cauchy problem, but I wouldn't call it local.

I have not discussed (and have no intention to discuss) locality/non-localty in non-relativistic field theories. So, why are we talking about the heat equation or other non-relativistic equations?

 

[Demystifier]

So, why are we talking about the heat equation or other non-relativistic equations?

Because some of us are interested in the concept of (non)locality in a wider context.

 

[vanhees71]

That's a well-known issue in relativistic transport theory (which is local by construction). When you go to the next coarse-grained level of description, you assume local thermal equilibrium and small deviations from it. In 0th order of the gradient expansion you get something like ideal relativistic fluid equations, which are ok. In the next order, linear in the gradients, you get something like the Navier-Stokes equations, which are unfortunately acausal (the same holds for heat conduction). You have to go at least to the next order. If you use the relaxation-time approximation of the Boltzmann equation you end up with Israel-Stewart viscous hydrodynamical equations, which are causal.

Today relativistic viscous fluid dynamics of higher orders are systematically derived from the transport equations via moment expansions. See, e.g.,

https://inspirehep.net/literature/1089847

 

[dx]

On a related note, in algebraic and axiomatic quantum field theory, one defines or characterises a quantum field theory by its algebra of local observables. See "local quantum physics" by Haag (https://www.springer.com/gp/book/9783642973062)