Sum over histories, including histories that violate conservation laws?

[bcrowell]

It's been 20 years since I took field theory in grad school, and I didn't really understand it all that well even then, so I'm basically looking for a very low-level explanation of the following issue...

In the sum-over-histories approach, there is the question of which histories to include in the sum and which to omit. I'm trying to understand how one decides what to include. If I'm understanding correctly, in many cases you want to omit histories that violate some principle of physics such as a conservation law. For instance, when you're drawing Feynman diagrams you don't normally include ones that violate conservation of charge. On the other hand, it may happen that a conservation law emerges only *after* you calculate the superposition. For example, if you're calculating a classical diffraction pattern, the total energy just comes out to be the right amount, and there's no way to impose energy conservation before summing. In QED, I seem to remember that one integrates over values of the variables that violate conservation of energy-momentum, and I suppose this might be fundamentally necessary because of the Heisenberg uncertainty principle...?

Can anyone tie this up in a nice package with a red ribbon on top for me? Is there any general way to understand why we would or would not sum over histories that violate conservation laws?

Thanks!

-Ben

 

[Ben Niehoff]

At every vertex in a Feynman diagram, we impose

1. Conservation of charge
2. Conservation of spin, and
3. Conservation of linear 4-momentum

Those are the usual most important conservation laws. The weirdness that you remember is probably that we do not enforce E = mc^2 when integrating over momenta in loops. That is, virtual particles are not required to be "on shell" (the mass shell; i.e., the hyperboloid in Minkowski space defined by $p^\mu p_\mu = -m^2$).

There are some other issues with conservation laws, anomaly currents, and Ward identities that I don't remember off the top of my head.

 

[tom.stoer]

Typically every conservation law which is due to a local gauge symmetry like U(1) for charge in QED, SU(3) for color-charge in QCD, energy-momentum for Poincare-invariance (one can formulate GR as a Poincare-gauge theory) MUST hold in the quantum theory as well. Gauge anomalies typically lead to inconsistent theories.

Conservation laws due to global symmetries (like the axial symmetry) can indeed be broken in the quantum theory; this is called "anomaly" - and is welcome in order to explain certain phenomena (the axial anomaly explains the large eta' mass = meson singulett which is much heavier than the meson octet states)

 

[Demystifier]

In the sum-over-histories approach, there is the question of which histories to include in the sum and which to omit. I'm trying to understand how one decides what to include. If I'm understanding correctly, in many cases you want to omit histories that violate some principle of physics such as a conservation law.

You misunderstood something. In the sum-over-histories approach, ALL conceivable histories are included.

For instance, when you're drawing Feynman diagrams you don't normally include ones that violate conservation of charge.

Feynman diagrams are NOT representations of histories in the sum-over-histories approach. In fact, Feynman diagrams have not much to do with the sum-over-histories approach at all.

 

[Demystifier]

Is there any general way to understand why we would or would not sum over histories that violate conservation laws?

Here is why we WOULD (which is a verbal description of essential steps in the derivation of the sum-over-histories approach):

At a given time you insert a unit operator, which is a sum over all possible values of the position (or momentum). At another time you do the same. The two expressions at different times are multiplied. The product of two sums is a sum of all possible products of two factors. A product of two different positions (or momenta) at different times corresponds to a history in which the particle changes the position (or momentum) accordingly. In this way, it is clear that any sequence of changing positions (or momenta) is possible, which corresponds to histories in which energy and/or momentum is not conserved.

Does it help?

 

[JesseM]

You misunderstood something. In the sum-over-histories approach, ALL conceivable histories are included.

Would this still be true in relativistic QM/field theory if you were trying to find the probabilities of detecting a particle at various positions/times, and that particle had been emitted at a known position and time? (say, because the emitter was covered by a shutter that had only opened briefly at a known time) In this case the probability of detecting the particle at a position/time that was outside the future light cone of the emission event would be zero, correct? If that's true would FTL histories be included?

 

[Demystifier]

Would this still be true in relativistic QM/field theory if ...

Yes it would.

In this case the probability of detecting the particle at a position/time that was outside the future light cone of the emission event would be zero, correct?

Correct.

If that's true would FTL histories be included?

Yes they would. They may be FTL at intermediate points.

 

[bcrowell]

You misunderstood something. In the sum-over-histories approach, ALL conceivable histories are included.

Hmm...I can conceive of some pretty crazy stuff :-) How about a history where an electron turns itself into a photon?

 

[Demystifier]

Hmm...I can conceive of some pretty crazy stuff :-) How about a history where an electron turns itself into a photon?

In quantum field theory, that is also one of the possibilities.

 

[bcrowell]

Hmm...I can conceive of some pretty crazy stuff :-) How about a history where an electron turns itself into a photon?

In quantum field theory, that is also one of the possibilities.

Are we imagining the same thing? I'm talking about a case where the world-line of the electron terminates, and both conservation of charge and conservation of angular momentum are violated.
In practice, you wouldn't bother summing over possibilities like that, would you? And supposing you did sum over ones like that -- would the amplitude for such a process cancel out to zero exactly?

 

[Lapidus]

Hmm...I can conceive of some pretty crazy stuff :-) How about a history where an electron turns itself into a photon?

Z(J)= path integral over all field configuration = exp(sum of all connected Feynman graphs)

Every Feynman graph you can conceive of is allowed. But by drawing those graphs, you have to stick to Feynman rules of the theory at hand.

 

[bcrowell]

Every Feynman graph you can conceive of is allowed. But by drawing those graphs, you have to stick to Feynman rules of the theory at hand.

So I guess to me the situation still seems messy. The motivation for the rules of the Feynman graphs involves conservation laws like conservation of charge and angular momentum. We impose these laws a priori, while others end up being obeyed only because the processes that violate them end up with zero total amplitude...?

 

[Ben Niehoff]

The Feynman rules come from taking functional derivatives of Z. Or ultimately, they come from the Lagrangian itself. The kinetic terms in the Lagrangian give you propagators, and the interaction terms give you vertices. In QED, the interaction term looks like

$$\bar{\psi} \gamma^\mu \psi A_\mu$$,

so the only vertex you can get is 1 photon line joined to 2 electron lines (interpreted as electrons or positrons, depending on the orientation). So charge conservation is automatic, but the reason it is automatic is because it was there in the Lagrangian to begin with.

 

[bcrowell]

The Feynman rules come from taking functional derivatives of Z. Or ultimately, they come from the Lagrangian itself. The kinetic terms in the Lagrangian give you propagators, and the interaction terms give you vertices. In QED, the interaction term looks like

$$\bar{\psi} \gamma^\mu \psi A_\mu$$,

so the only vertex you can get is 1 photon line joined to 2 electron lines (interpreted as electrons or positrons, depending on the orientation). So charge conservation is automatic, but the reason it is automatic is because it was there in the Lagrangian to begin with.

OK, that makes sense, but to me it doesn't really seem to make the picture any tidier. It just means that we attribute the a priori laws to the structure of the Lagrangian, whereas the other ones emerge because amplitudes cancel.

Are there cases where you can *either* impose a rule a priori *or* have it emerge because the amplitudes cancel, so that imposing the rule is just a matter of computational convenience? For instance, if I'm trying to derive Snell's law in classical optics from the principle of least time, I don't really have to assume a priori that the light rays refract only at the interface, but the computation is a lot easier and simpler if I do.

 

[Ben Niehoff]

OK, that makes sense, but to me it doesn't really seem to make the picture any tidier. It just means that we attribute the a priori laws to the structure of the Lagrangian, whereas the other ones emerge because amplitudes cancel.

Well, yes. From Noether's theorem, the structure of the Lagrangian gives us certain conservation laws. For Lorentz-invariant Lagrangians, energy is not a Noether current; only the combination $p_\mu$ is.

Are there cases where you can *either* impose a rule a priori *or* have it emerge because the amplitudes cancel, so that imposing the rule is just a matter of computational convenience? For instance, if I'm trying to derive Snell's law in classical optics from the principle of least time, I don't really have to assume a priori that the light rays refract only at the interface, but the computation is a lot easier and simpler if I do.

I'm not sure. I know in QED calculations, we sometimes choose the gauge $A_0 = 0$, which is obviously not Lorentz-covariant. At the end of the calculation, it takes some work to show that the answer is in fact Lorentz-covariant after all.

Technically, when you take functional derivatives of Z, you take all possible combinations of them. But most of those combinations will vanish identically, except for the ones that correspond to terms in the Lagrangian. I'm not sure if this should be interpreted as summing over impossible histories and having them cancel, or simply as a shortcut to eliminate them at the outset.

 

[Demystifier]

Are we imagining the same thing? I'm talking about a case where the world-line of the electron terminates ...

We don't. In the path integral formulation of QFT, there are no particle world lines. Only field configurations.

 

[tom.stoer]

All reasoning based on path integrals should be possible w/o ever referring to Feynman diagrams which are only calculational or book-keeping tools in the perturbative domain. The path integral (action, measure, boundary configuration ~ typically vacuum) itself can be taken as definiton of the quantum mechanical theory (or quantum field theory). The action respects certain (classical symmetries); if the full regularized path integral including the measure repects these symmetries as well, then the quantum theory is free of quantization anomalies which means that all classical symmetries following from the action are respected.

We started with the question whether there is a "general way to understand why we would or would not sum over histories that violate conservation laws?".

The explanation based on the path integral is that we do not sum over all configurations but only over configurations generated by the path integral, that means configurations respecting certain symmetries (Lorentz invariance, gauge invariance - even when we fix a gauge e.g. A°=0 which does neither break gauge nor Lorentz invariance) and therefore the corresponding conservation laws (energy-momentum-conservation, charge conservation).

The path integral (espacially in QFT) is a formal object, but it can be evaluated in certain approximations. Only in some approximations it makes sense to attribute entities like energy to intermediate "steps on a path". This is done in the Feynman path integral and one finds that energy-momentum conservation and charge conservation holds at the vertices. It is not the case that it is violated between the vertices (on the internal lines) but instead there is no reasonable notion of energy and momentum between the vertices. There are integration variables denoted by p, but i wouldn't say that one can point towards a certain point in a Feynman loop and ask for energy and momentum at this pont; this question does not make sense.

Fact is that whenever it's possible to (reasonably) ask for the value of a certain quantity like energy, momentum, charge etc., then one finds that the quantity is conserved i.e. agrees with the value for the boundary configuration, typically the "in"-state.