# Locality in field theory

• I
In quantum field theory (QFT) from what I've read locality is the condition that the Lagrangian density ##\mathscr{L}## is a functional of a field (or fields) and a finite number of its (their) spatial and temporal derivatives evaluated at a single spacetime point ##x^{\mu}=(t,\mathbf{x})##, i.e. it should be of the form $$\mathscr{L}=\mathscr{L}\left(\phi(t,\mathbf{x}),\frac{\partial\phi(t,\mathbf{x})}{\partial t},\frac{\partial\phi(t,\mathbf{x})}{\partial x^{i}}\right)$$ but why is this required?

Is it simply that we require physics to be local, i.e. the physical state of a system at particular point should only depend on what's "going on" in the immediate neighbourhood of that point. As one can reconstruct the behaviour of the fields in the neighbourhood of a spacetime point given the values of the field and a finite number of derivatives at that point (via a Taylor expansion), this leads us to require that the Lagrangian density should only depend on the value of the field and a finite number of its derivatives a that point?

Also, is it to impose that no two fields should be able to interact directly with one another if they are located at different spacetime points (however small their separation) since this would constitute action at a distance. As such, no interaction terms of the form ##\phi(x)\phi(y)##, where ##x^{\mu}\neq y^{\mu}## are allowed since otherwise this would imply that fields at different points in space and time could directly influence, with nothing mediating such an interaction.

Its actually the cluster decomposition property:

Thanks
Bill

But why does this require that Lagrangian densities should depend only on fields and (a finite number of) their derivatives at a single spacetime point? Furthermore, why does it require that interactions should occur at single spacetime points? Intuitively, I thought that it was because fields at different spacetime points should not be able to influence one another directly - there should be some mediator that mediates any interaction such that they can interact indirectly. If interactions didn't occur at single spacetime points then there would be a reference frame in which causality is violated (i.e. superluminal interaction).

bhobba
Mentor
But why does this require that Lagrangian densities should depend only on fields and (a finite number of) their derivatives at a single spacetime point? Furthermore, why does it require that interactions should occur at single spacetime points? Intuitively, I thought that it was because fields at different spacetime points should not be able to influence one another directly - there should be some mediator that mediates any interaction such that they can interact indirectly. If interactions didn't occur at single spacetime points then there would be a reference frame in which causality is violated (i.e. superluminal interaction).

That would require study of Wienbergs book. I have a copy but its beyond my current level - maybe when I get the time.

Thanks
Bill

stevendaryl
Staff Emeritus
In quantum field theory (QFT) from what I've read locality is the condition that the Lagrangian density ##\mathscr{L}## is a functional of a field (or fields) and a finite number of its (their) spatial and temporal derivatives evaluated at a single spacetime point ##x^{\mu}=(t,\mathbf{x})##, i.e. it should be of the form $$\mathscr{L}=\mathscr{L}\left(\phi(t,\mathbf{x}),\frac{\partial\phi(t,\mathbf{x})}{\partial t},\frac{\partial\phi(t,\mathbf{x})}{\partial x^{i}}\right)$$ but why is this required?

Is it simply that we require physics to be local, i.e. the physical state of a system at particular point should only depend on what's "going on" in the immediate neighbourhood of that point. As one can reconstruct the behaviour of the fields in the neighbourhood of a spacetime point given the values of the field and a finite number of derivatives at that point (via a Taylor expansion), this leads us to require that the Lagrangian density should only depend on the value of the field and a finite number of its derivatives a that point?

Also, is it to impose that no two fields should be able to interact directly with one another if they are located at different spacetime points (however small their separation) since this would constitute action at a distance. As such, no interaction terms of the form ##\phi(x)\phi(y)##, where ##x^{\mu}\neq y^{\mu}## are allowed since otherwise this would imply that fields at different points in space and time could directly influence, with nothing mediating such an interaction.

This is a little off-topic, but years ago, John Baez had a short post explaining that string theory is actually nonlocal in this sense, but I can't remember why that's the case.

A. Neumaier
Its actually the cluster decomposition property:
Locality is not cluster decomposition itself but what makes cluster decomposition easy to prove. (There is a particle-oriented non-Lagrangian version of relativistic quantum mechanics that implements the cluster decomposition principle, but this is far less developed and not taught in the standard courses. See the section ''Is there a multiparticle relativistic quantum mechanics?'' from Chapter B1: The Poincare group of my theoretical physics FAQ.)

More precisely one needs local interactions and causal commutation relations. It is the latter that ensure that a field can be prepared independently at an arbitrary finite number of mutually spacelike points. This is very well explained in Weinberg's first volume on quantum field theory.

Locality is the quantum equivalent of a classical dynamics given by a conservative Poincare-invariant system of hyperbolic partial differential equations. This ensures the causal and relativistically consistent behavior of its predictions. No other way is known to implement the latter in a quantum field theory.

Effective field theories are strictly speaking always nonlocal but are truncated to a local field theory to make them tractable. Conceptually, nonlocality can always been thought of as having a memory distributed in space and time, which can be eliminated by introducing additional degrees of freedom that encode the memory in the here and now. This is why a fundamental quantum field theory is believed to be necessarily local. (On the other hand, string theory, possibly even more fundamental, is intrinsically nonlocal since strings are extended, but it is not a quantum field theory.)

• bhobba
Effective field theories are strictly speaking always nonlocal but are truncated to a local field theory to make them tractable. Conceptually, nonlocality can always been thought of as having a memory distributed in space and time, which can be eliminated by introducing additional degrees of freedom that encode the memory in the here and now. This is why a fundamental quantum field theory is believed to be necessarily local. (On the other hand, string theory, possibly even more fundamental, is intrinsically nonlocal since strings are extended, but it is not a quantum field theory.)

Intuitively then, can one think of locality in terms of the information needed to describe a physical system at a point, i.e. to describe the physics of a system at a point one should only need information about the behaviour of the system arbitrarily close to that point, which is encoded in the values of the fields and their derivatives at that point?

In essence, to describe a physical system at point ##x^{\mu}## one should not have to know its behaviour at some other arbitrary point ##y^{\mu}##, right?!

The formal requirement in relativistic quantum field theory is that observables at spacelike separation commute.

That requirement is separate from the locality you mention, which is a matter of convenience

What so terms of the form ##\phi(x)\phi(y)## don't appear in Lagrangian densities purely because of convenience? Isn't there some physical motivation behind it, such as a field at ##x^{\mu}## should not be able to directly interact with a field at ##y^{\mu}## (for ##x^{\mu}\neq y^{\mu}##)?

A. Neumaier
locality you mention, which is a matter of convenience
not completely, as the Aharanov-Bohm effect mentioned by Matthew Schwartz shows.

A. Neumaier
we require physics to be local
Not generally. In the nonrelativistic quantum field theory of charged particles one has nonlocal Coulomb interactions.

not completely, as the Aharanov-Bohm effect mentioned by Matthew Schwartz shows.

Ok. I thought locality was an original motivation to avoid "mysterious" action-at-a-distance, in which no medium is required to "carry" interactions from point to point?! By requiring locality one can guarantee that interactions are causal and also construct theories which match intuition and observation - that physical systems can not directly influence one another without something mediating the interaction between them (if this weren't the case then, for example, in the classical case, one could move an object without actually being in physical, direct contact with it).

A. Neumaier
for example, in the classical case, one could move an object without actually being in physical, direct contact with it)
This is indeed the case with magnetically supported trains. Of course we think that the space between the rails and the train is not empty but filled with a magnetic field.

Of course we think that the space between the rails and the train is not empty but filled with a magnetic field.

That's what I mean. The magnetic force that causes the train to levitate is mediated by the magnetic field - interactions occur at points along a path between the magnet and the train.

samalkhaiat
In quantum field theory (QFT) from what I've read locality is the condition that the Lagrangian density ##\mathscr{L}## is a functional of a field (or fields) and a finite number of its (their) spatial and temporal derivatives evaluated at a single spacetime point ##x^{\mu}=(t,\mathbf{x})##, i.e. it should be of the form $$\mathscr{L}=\mathscr{L}\left(\phi(t,\mathbf{x}),\frac{\partial\phi(t,\mathbf{x})}{\partial t},\frac{\partial\phi(t,\mathbf{x})}{\partial x^{i}}\right)$$ but why is this required?

Is it simply that we require physics to be local, i.e. the physical state of a system at particular point should only depend on what's "going on" in the immediate neighbourhood of that point. As one can reconstruct the behaviour of the fields in the neighbourhood of a spacetime point given the values of the field and a finite number of derivatives at that point (via a Taylor expansion), this leads us to require that the Lagrangian density should only depend on the value of the field and a finite number of its derivatives a that point?

Also, is it to impose that no two fields should be able to interact directly with one another if they are located at different spacetime points (however small their separation) since this would constitute action at a distance. As such, no interaction terms of the form ##\phi(x)\phi(y)##, where ##x^{\mu}\neq y^{\mu}## are allowed since otherwise this would imply that fields at different points in space and time could directly influence, with nothing mediating such an interaction.

Your understanding is correct. So, you don’t need to confuse yourself. If you can get hold of Bjorken & Drell’s text, Vol 2 “Relativistic Quantum Fields”, read section 11.1: “Implications of a Description in terms of Local Fields” page 3-5. These are the only 2 pages worth reading in that textbook.

• vanhees71 and bhobba
Your understanding is correct. So, you don’t need to confuse yourself. If you can get hold of Bjorken & Drell’s text, Vol 2 “Relativistic Quantum Fields”, read section 11.1: “Implications of a Description in terms of Local Fields” page 3-5. These are the only 2 pages worth reading in that textbook.

Oh good. To be honest, I was getting myself in a right muddle trying to figure out what I was missing!
I found a copy of volume 1 of Bjorken and Drell's, but I haven't been able to find a copy of volume 2 as of yet unfortunately.

atyy
Your understanding is correct. So, you don’t need to confuse yourself. If you can get hold of Bjorken & Drell’s text, Vol 2 “Relativistic Quantum Fields”, read section 11.1: “Implications of a Description in terms of Local Fields” page 3-5. These are the only 2 pages worth reading in that textbook.

But what about gauge theories, where the gauge-invariant object is something nonlocal like a Wilson loop?

DrDu
A. Neumaier
But what about gauge theories, where the gauge-invariant object is something nonlocal like a Wilson loop?
In QED (abelian gauge field theory), field strengths are local operators, but gauge invariant charged operators are nonlocal.

In the local formulation of gauge theories there is a local field algebra of not necessarily gauge invariant operators and gauge invariance is implemented through gauge invariant states on the local field algebra. The local formulation is essential in all covariant formulations.

• bhobba and vanhees71
atyy
In QED (abelian gauge field theory), field strengths are local operators, but gauge invariant charged operators are nonlocal.

In the local formulation of gauge theories there is a local field algebra of not necessarily gauge invariant operators and gauge invariance is implemented through gauge invariant states on the local field algebra. The local formulation is essential in all covariant formulations.

But what about the nonlocal example in Eq 9.97 - Eq 9.99 of http://isites.harvard.edu/fs/docs/icb.topic521209.files/QFT-Schwartz.pdf ?

samalkhaiat
But what about gauge theories, where the gauge-invariant object is something nonlocal like a Wilson loop?

I don’t understand what you are saying. I also don’t know how much you know about the subject. So, I will not be technical. However, if you include some mathematics into your posts, then I can be as technical as you want me to be.
1) The program of regularization and renormalization has only been successfully carried out in local covariant theories. Local, manifestly covariant formulation is by far the best way to handle the UV-structure of the theory. And that exactly is the success story of QFT.
2) In gauge field theories, being Hemitian and gauge invariant is not sufficient to qualify as an observable. In addition, microscopic causality strongly forces us to deal with local operators, i.e., operators having compact support in some space-like region.
3) Gauge field theories are local theories. This fact does not change when we reformulate gauge theories in terms of Wilson loop variables.
4) The gauge invariant Wilson loops are non-local and over-complete relative to the local gauge potential. They are therefore constrained variables. Indeed, the generalized Mandelstam constraints form a sufficient algebraic set of conditions on Wilson loop variables and guarantee that the Wilson loop variables describe a local gauge theory.

A. Neumaier
But what about the nonlocal example in Eq 9.97 - Eq 9.99 of http://isites.harvard.edu/fs/docs/icb.topic521209.files/QFT-Schwartz.pdf ?
By eliminating a field in a local formulation, Schwartz obtains a reduced nonlocal description, which upon expansion and approximation would become an effective field theory. This is just a very special way in which the reduced descrption can be given explicitly rather than only perturbatively. But the natural (and hence - as Schwartz mentions - in terms of computability much more flexible) representation is the local one.

atyy
By eliminating a field in a local formulation, Schwartz obtains a reduced nonlocal description, which upon expansion and approximation would become an effective field theory. This is just a very special way in which the reduced descrption can be given explicitly rather than only perturbatively. But the natural (and hence - as Schwartz mentions - in terms of computability much more flexible) representation is the local one.

Yes, I agree the local formulation is the more natural one. However, what I understand Schwartz to be saying is that it is not necessary.

A. Neumaier
Yes, I agree the local formulation is the more natural one. However, what I understand Schwartz to be saying is that it is not necessary.
This is like making the local Maxwell's equations nonlocal by rewriting them in terms of the retarded Lienard-Wiechert potential. it gives an equivalent description. But the latter would never be regarded as a starting point, since it begs for explanation - and the explanation is through the local description.

DrDu
atyy
Isn't that just the mechanism behind global gauge symmetry breaking? I.e. the Goldstone boson ##\pi## getting eaten by A in a special gauge leading to a massive vector potential which appears to break Gauge symmetry?

I don't think so. In Eq 9.98, the action is already for a massive field, but written in a local form (where locality is defined as in the OP). Then in Eq 9.99, he rewrites the same action in a nonlocal way that (at least classically) should have the same physics.

DrDu
I don't think so. In Eq 9.98, the action is already for a massive field, but written in a local form (where locality is defined as in the OP). Then in Eq 9.99, he rewrites the same action in a nonlocal way that (at least classically) should have the same physics.
Ok, I found this:
https://en.wikipedia.org/wiki/Stueckelberg_action

• atyy
vanhees71
Gold Member
2021 Award
Well, if you integrate out fields in a local QFT you end up with a non-local effective theory. On the other hand, if you quantize gauge theories in the operator formalism, which is a quite tricky business, you learn that the gauge fixing conditions cannot applied on the operator level but only on the level of states, leading to a formalism that was introduced for QED in the 1950ies by Gupta and Bleuler. For a nice introduction, see

Nachtmann, O.: Elementary Particle Physics - Concepts and Phenomenology, Springer-Verlag, 1990

For the non-Abelian case it's pretty much more complicated and relies on BRST symmetry of the gauge-fixed action as it was derived within the path-integral formulation in terms of the Faddeev-Popov approach. This you find presented in all details in the following three papers

Kugo, T., Ojima, I.: Manifestly Covariant Canonical Formulation of the Yang-Mills Field Theories. I, Progress of Theoretical Physics 60(6), 1869–1889, 1978
http://dx.doi.org/10.1143/PTP.60.1869

Kugo, T., Ojima, O.: Manifestly Covariant Canonical Formulation of Yang-Mills Field Theories. II: SU (2) Higgs-Kibble Model with Spontaneous Symmetry Breaking, Progress of theoretical physics 61(1), 294–314, 1979
http://dx.doi.org/10.1143/PTP.61.294

Kugo, T., Ojima, I.: Manifestly Covariant Canonical Formulation of Yang-Mills Field Theories. III—Pure Yang-Mills Theories without Spontaneous Symmetry Breaking, Progress of Theoretical Physics 61, 644–655, 1979
http://dx.doi.org/10.1143/PTP.61.644

DrDu
Yes, I agree the local formulation is the more natural one. However, what I understand Schwartz to be saying is that it is not necessary.
From what I understood now is that the example given by Schwarz implies a non-compact gauge group, while the gauge theories used in field theory are compact ones. This is physically reasonable, as the reason we really need a gauge theory is the Aharonov-Bohm effect, which relies on the gauge group being compact.
E.g. the line integral over the vector potential gives the encompassed magnetic flux and we can't gauge it away.

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• atyy
vanhees71
Gold Member
2021 Award
I think what Schwartz discusses there is the realization of an Abelian massive vector field as a gauge field. This is a remarkable model, because it shows that in the Abelian case, i.e., gauge group U(1), you can formulate a gauge-symmetric renormalizable model with massive gauge bosons without the Higgs mechanism. This construct does not work for the non-Abelian case. There you need the Higgs mechanism to consistently describe massive gauge bosons and/or fermions for chiral gauge groups as in the electroweak sector of the Standard Model.

• atyy
DrDu
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