Understanding Local and Nonlocal Operators in Quantum Field Theory

[Demystifier]

How do you phrase all that in terms of distributions?

I don't, I leave that to mathematicians.

 

[Nullstein]

I think it's in fact quite physical that a field at one point depends on other fields at other points. For instance, static electric field at one point depends on static charge distribution at all other points.

But the field configuration is modeled by the quantum state. The field operator $\phi(x)$ just probes the field configuration at a specific point. The analog in quantum mechanics would be the position operators $x_i$. E.g., we don't want $x_1$ to probe the position of particle $i=2$. The analog of $\eta(x)$ would be something like the center of mass position $\frac{1}{2}(x_1 + x_2)$.

 

[Demystifier]

But the field configuration is modeled by the quantum state. The field operator $\phi(x)$ just probes the field configuration at a specific point. The analog in quantum mechanics would be the position operators $x_i$. E.g., we don't want $x_1$ to probe the position of particle $i=2$. The analog of $\eta(x)$ would be something like the center of mass position $\frac{1}{2}(x_1 + x_2)$.

Those are good analogies. But if you take into account interactions between particles, there is nothing strange with the idea that the fist particle probes the position of the second particle. In fact, in the absence of such probing, physical measurement would be impossible. Besides, the center of mass position behaves very much like a point particle (it moves as if all mass was concentrated at that point), despite the fact that this point may not contain matter at all.

You might want to study only kinematics (not the dynamics), in which case you don't want to know how one particle depends on other particles or how the field at one point depends on fields at other points. And you might want to study only fundamental degrees of freedom, in which case you are not interested in effective degrees such as the center of mass position. You might restrict your thinking in that way, but this may be misleading. Especially if all field theories are just effective theories after all.

So I would suggest to think of local field as something that formally behaves as a local field, even if this is not a true local field at the fundamental level. That's completely analogous to the fact that the center of mass position formally behaves as a pointlike particle, even if this is not a true pointlike particle at the fundamental level.

 

[samalkhaiat]

What a mess! Okay, I am going to settle this (local thingy) once and for all. I will start by explaining the term in the relativistic classical field theory and then in QFT.

Classical Field Theory: Here the field variables $\phi_{a}(x)$ are continuous functions on $\mathbb{R}^{(1,3)}$. The field equations and their invariants are obtained from the Lagrangian function $\mathcal{L}(x)$ which is (quasi) invariant under the action of the Poincare group. Usually $\mathcal{L}(x)$ is taken to be a real function of the fields $\phi_{a}(x)$ and of their first derivatives $\partial_{\mu}\phi_{a}$ leading (in this case) to field equations being differential equations of the second order. If $\mathcal{L}(x)$ depends on the state of the fields in an infinitesimally small neighbourhood of the point $x$, i.e., on the values of $\phi_{a}$ and of a finite number of their partial derivatives evaluated at the point $x$, then $$\mathcal{L}(x) = \mathcal{L}\left(\phi_{a}(x) , \partial_{\mu}\phi_{a}(x) \right) ,$$ is called local Lagrangian, and the corresponding theory is said to be a local field theory. In here, the term “local fields” are also used simply because the changes in the fields at a point $x$ are determined by the properties of the fields infinitesimally close to the point $x$, i.e., the dynamical evolution is completely determined by the initial data ($\phi (0, \vec{x}) , \partial_{t}\phi (0 , \vec{x})$) and the field equation.

Exercise (1): You are given the Cauchy surface at $t_{0}$, $$\phi (t_{0}, \vec{x}) = F (\vec{x}), \ \ \ \partial_{t}\phi (t_{0} , \vec{x}) = G (\vec{x}),$$ and the field equation $$\frac{\partial^{2}\phi}{\partial t^{2}} = \nabla^{2} \phi - m^{2} \phi .$$ You are asked to reconstruct the dynamical evolution of the Klein-Gordon field.

In the opposite case when, for example, the Lagrangian has the form $$\mathcal{L}(x) = \int d^{4}y \ \Lambda (\phi (x) , \phi (y) , \partial \phi (x) , \partial \phi (y)) ,$$ one obtains so-called nonlocal theories which are plagued with incurable diseases.

The invariance of the local theory under continuous (symmetry) group allows us to construct an object (for example 4-vector current $j^{\mu}(x)$), which is a local function of $\phi_{a}(x)$ and $\partial_{\mu}\phi_{a}(x)$, satisfying local conservation law $\partial_{\mu}j^{\mu}(x) = 0$ which leads to a time-independent (constant) quantity $Q = \int d^{3}x \ j^{0}(t, \vec{x})$ and is called a global charge. In here, there are two reasons for adjective “global” : (1) because $\frac{dQ}{dt} = 0$ and the fact that it is defined by integrating out the $\vec{x}$-dependence of $j^{0}(t , \vec{x})$, but most importantly (2) because one can show that the time-independent $Q$ generates the global symmetry group of the theory.

Quantum Field Theory: In QFT the field $\varphi (x)$ becomes an operator-valued distribution on $\mathbb{R}^{(1,3)}$. This quantum field is related to an operator on an abstract Hilbert space $\mathcal{H}$ and, in contrast to the classical field, ceases to describe the state of the system; this state is now represented by an abstract vector of the Hilbert space. It is clear why we need operators in the quantum theory, but why operator-valued distributions and not operator-valued (continuous) functions on $\mathbb{R}^{(1,3)}$?
That is to say that we need to understand why the quantum field, in contrast to the classical one, cannot be a continuous function in all 4 variables $(x^{0} , \vec{x})$. The reason is that the (free) field equation alone does not give the full characteristic of the (free) field, the missing parts are the equal-time commutation relations which, indeed, play the role of that of the initial data in the classical case. To see that, consider the simple case of KG field equation $$(\partial^{2} + m^{2}) \varphi (x), \ \ \ \ \ (1)$$ together with the equal-time commutation relations $$[ \varphi (t , \vec{x}) , \varphi (t , \vec{y})] = 0, \ \ \ \ \ \ \ \ \ \ \ (2a)$$$$[ \varphi (t , \vec{x}) , \dot{\varphi} (t , \vec{y})] = i \delta^{3} (\vec{x} - \vec{y}). \ \ \ \ (2b)$$ Now, look at the non-equal-time commutator $$[\varphi (x) , \varphi (y)] = i\Delta (x,y) \ \mbox{id}.$$ Since, $\varphi (x)$ satisfies the KG equation (1), it follows that $$(\partial^{2}_{x} + m^{2}) \Delta (x,y) = 0.$$ And from (2) we obtain the initial conditions on $\Delta$ $$\Delta (x,y)|_{x^{0} = y^{0} = t} = 0,$$$$\frac{\partial \Delta (x,y)}{\partial x^{0}}|_{x^{0} = y^{0} = t} = - \delta^{3} (\vec{x} - \vec{y}) .$$ Since these initial conditions depend only on $(\vec{x} - \vec{y})$, $x^{0} - y^{0} = 0$ and are distributions, the same must be true for the solution $\Delta (x,y)$.
In QFT there is a well-founded procedure in which the field distribution can be smeared out with a “good” function $f(x)$ in such a way that it become, in general unbounded, operator acting in the Hilbert space and is linear on the space of “good” functions: $$\varphi (f) \equiv \int d^{4}x \ \varphi (x) f(x) .$$The operators $\varphi (f)$ for all $f$ have a common dense domain of definition $\Omega \subset \mathcal{H}$ and we want $\varphi (f): \Omega \to \Omega$, viz. $$\varphi (f) \Omega \subset \Omega .$$ The domain $\Omega$ must also be stable under the action of the infinite-dimensional unitary representation $U(a ,\Lambda )$ of the Poincare group. Further, the linear functional $$f \mapsto \langle \Psi |\varphi (f) |\Phi \rangle ,$$ should be continuous for any $|\Psi \rangle , |\Phi \rangle \in \mathcal{H}$ with respect to the topology of the spaces of “good” functions. These spaces are usually taken to be $\mathcal{S}(\mathbb{R}^{4})$, the space of rapidly decreasing functions in $\mathbb{R}^{4}$, or $\mathcal{D}(\mathcal{O})$, the space of compactly-supported functions on the (bounded) open set $\mathcal{O} \subseteq \mathbb{R}^{4}$.
Now, we come to the badly discussed issue of this thread, so please pay careful attention to the following: In the case that $\mathcal{O} \subset \mathbb{R}^{4}$ is a finite spacetime region, we call the operator $$\varphi (f) = \int d^{4}x \ \varphi (x) \ f(x) , \ \ \ f \in \mathcal{D}(\mathcal{O}) ,$$ a smeared local operator if it satisfies the following axioms:
i) Poincare covariance, $$U^{\dagger}(a, \Lambda) \varphi_{r}(f)U(a , \Lambda) = D_{r}{}^{s}(\Lambda) \varphi_{s} (f_{(a , \Lambda)}),$$ where $f_{(a , \Lambda)}(x) = f\left(\Lambda^{-1}(x - a)\right)$, and $D(\Lambda)$ is a finite-dimensional representation of the proper Lorentz group $SO^{\uparrow}(1,3)$. In the particular case of translation $U(a , 1) = e^{i a^{\mu}P_{\mu}}$, we have $$e^{-i a^{\mu}P_{\mu}} \ \varphi (f) \ e^{i a^{\mu}P_{\mu}} = \varphi (f_{(a , 1)}) , \ \ \ f_{(a,1)} (x) = f(x - a).$$ The infinitesimal version of this reads $$[iP_{\mu} , \varphi (f)] = \varphi (\partial_{\mu}f).$$
ii) Local (anti)commutativity $$[\varphi_{r}(f) , \varphi_{s} (g)] = 0,$$ whenever $\mbox{supp} \ f \in \mathcal{D}(\mathcal{O})$ is spacelike with respect to $\mbox{supp} \ g \in \mathcal{D}(\mathcal{O})$.
The existence of the unique vacuum together with axiom (ii) allow us to prove the separating property of the vacuum with respect to smeared local fields: since the causal complement for a bounded open set $\mathcal{O}$ is not empty, any smeared local operator $\varphi (f)$, $\forall f \in \mathcal{D}(\mathcal{O})$ annihilating the vacuum is vanishing in itself, i.e., $\varphi (f) |0 \rangle = 0 \ \ \Rightarrow \ \varphi (f) = 0$.

So, a non-local field (whatever it is) is one that does not satisfy the above two axioms and, therefore, with respect to it the separating property of the vacuum does not hold. Also, the absence of $x$ in the arguments of a field (i.e., the smearing procedure) does not make the field non-local.

Exercise (2): Recall the local conservation law in classical field theory $\partial^{\mu} j_{\mu} = 0$. In QFT the vector current $j_{\mu}(x)$ becomes an operator-valued distribution. Let $f_{T} \in \mathcal{D}(\mathbb{R}), \ f_{R} \in \mathcal{D}(\mathbb{R}^{3})$ be test functions such that $\int dx^{0} f_{T}(x^{0}) = 1$, $f_{R}(\vec{x}) = 1$ for $|\vec{x}|\leq R$ and $f_{R}(\vec{x}) = 0$ for $|x| \geq 2R$. Write down the distributional conservation law $\partial^{0}j_{0}(x) + \sum_{k}^{3} \partial^{k}j_{k}(x) = 0$ in terms of smeared out local fields.

 

[Demystifier]

What a mess! Okay, I am going to settle this (local thingy) once and for all.

You destroyed all the fun, now we have nothing to discuss any more.

Now seriously! What book or review would you recommend on this stuff?

 

[vanhees71]

Great summary, but let's see, whether this helps to settle the issue .

 

[martinbn]

Great summary, but let's see, whether this helps to settle the issue .

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

 

[MathematicalPhysicist]

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

 

[MathematicalPhysicist]

You destroyed all the fun, now we have nothing to discuss any more.

Now seriously! What book or review would you recommend on this stuff?

I once tried reading a copy of Baez's textbook on Intro to algebraic and constructive QFT:
https://math.ucr.edu/home/baez/bsz_new.pdf
But I've seen that I first need to grasp Lie algebras and Lie Groups.
I don't know but I tend to forget all the details of the definitions unless I am using them repeatedly.

 

[samalkhaiat]

You destroyed all the fun, now we have nothing to discuss any more.

Hehehe, the destruction would have been greater if the terms quasi-local, relatively local and/or localized fields were included in the discussions.

Now seriously! What book or review would you recommend on this stuff?

As for books, I liked and learned a lot from R. Jost’s book “The General Theory of Quantized Fields”,(Amer. Math. Soc. Publication, 1963). There is also Haag’s book “Local Quantum Physics”, (Springer, 1996).
For review articles, see
Kastler, Robinson & Swieca, Comm. Math. Phys., 2, 108 (1966).
Ferrari & Picasso, Comm. Math. Phys., 35, 25 (1974),
and the all times classic review article of Claudio Orzalesi in Rev. Mod. Phys. 42, 381-409(1970).

 

[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)