Microcausality in algebraic QFT

[hellfire]

The condition of microcausality (commuting fields for spatially separated points) can be shown to hold in the Fock representation in quantum field theory (see e.g. Peskin & Schroeder section 2.4). However, in algebraic quantum field theory the condition of microcausality is postulated as an axiom (Haag-Kastler axioms). I had expected that a theory of observables that is defined in Minwkowski space-time should inherit naturally the causal structure of special relativity, and, therefore, microcausality should be a derivable result, but not an axiom. The fact that the Fock representation makes it possible to derive microcausality seems to be irrelevant (or of no generality) in algebraic quantum field theory due to the fact that there exist inequivalent representations of the algebra of observables. Is this picture correct? If yes, why is there a need to postulate microcausality in AQFT in contrast to the usual Fock representation in QFT?

 

[Hurkyl]

Consider this counterexample:

Let A be any C*-algebra that isn't commutative. Try to define an AQFT as follows:

$\mathcal{A}(U) = A$ for any open set U of Minkowski space.

For $U \subseteq V$, the embeddings $\mathcal{A}(U) \to \mathcal{A}(V)$ are identity maps.

The Poincaré group acts trivially on $\mathcal{A}(M)$.



It's easy to check that $\mathcal{A}$ is a functor, preserves monomorphisms, the Poincaré group acts continuously, and this even satisfies primitive causality¹. However, $\mathcal{A}$ fails to have the spacelike commutativity property.


Since spacelike commutativity is a desirable property that we wish to study, and this property is not a consequence of the other axioms, it follows that we need to include it (or something equivalent to it) in our definition of an AQFT.



(Disclaimer: I'm not a practicioner of quantum physics)


1: I'm going by the terminology on wikipedia.

 

[Haelfix]

It depends on the axiom system you use. There are AQFT systems that retrieve that particular result as a theorem. It just proved to be a physically reasonable starting point historically.

Even in regular physics, it need not be a theorem, and can be set up as an axiom (see Weinbergs approach vol 1).

 

[strangerep]

The condition of microcausality (commuting fields for spatially separated points) can be shown to hold in the Fock representation in quantum field theory (see e.g. Peskin & Schroeder section 2.4). However, in algebraic quantum field theory the condition of microcausality is postulated as an axiom (Haag-Kastler axioms). I had expected that a theory of observables that is defined in Minwkowski space-time should inherit naturally the causal structure of special relativity, and, therefore, microcausality should be a derivable result, but not an axiom. The fact that the Fock representation makes it possible to derive microcausality seems to be irrelevant (or of no generality) in algebraic quantum field theory due to the fact that there exist inequivalent representations of the algebra of observables. Is this picture correct? If yes, why is there a need to postulate microcausality in AQFT in contrast to the usual Fock representation in QFT?

I don't think one can "derive" microcausality from the Fock representation. Rather, one
just extracts what one has already put in earlier without realizing it. The Fock representation
is just an infinite-dimensional representation of the Poincare group for $m^2\ge 0$
and $E>0$. Microcausality is nothing more than the former assumption.

Many expositions of Wignerian classification of the unitary irreducible
representations of the Poincare group restrict themselves to $m^2\ge 0$ quite
early, and forget about the tachyonic representations $m^2<0$. (In fact, the
little group associated with tachyonic representations is very different from the
usual SO(3) little group we're familiar with for $m^2>0$.)

 

[hellfire]

Thanks for the insightful comments.

It's easy to check that $\mathcal{A}$ is a functor, preserves monomorphisms, the Poincaré group acts continuously, and this even satisfies primitive causality¹.

What does primitive causality mean? I do not understand what is written in wikipedia (btw what is meant there with "causal complement"?)

 

[Hurkyl]

What does primitive causality mean?

If V is an open subset of space-time, and $\bar{V}$ is the region that it causally determines, then the inclusion $\mathcal{A}(V) \to \mathcal{A}(\bar{V})$ is an isomorphism. i.e. any observable in $\bar{V}$ can be rewritten as an observable in V.

In particular, this extends to say that if U causally determines V (e.g. U might contain a slice of V's past or future lightcone), then any observable in V can be rewritten as an observable in U.

I do not understand what is written in wikipedia (btw what is meant there with "causal complement"?)

The causal complement of U is, I think, the region whose points are all spacelike separated from everything in U.

 

[hellfire]

I don't think one can "derive" microcausality from the Fock representation. Rather, one
just extracts what one has already put in earlier without realizing it. The Fock representation
is just an infinite-dimensional representation of the Poincare group for $m^2\ge 0$
and $E>0$. Microcausality is nothing more than the former assumption.

Many expositions of Wignerian classification of the unitary irreducible
representations of the Poincare group restrict themselves to $m^2\ge 0$ quite
early, and forget about the tachyonic representations $m^2<0$. (In fact, the
little group associated with tachyonic representations is very different from the
usual SO(3) little group we're familiar with for $m^2>0$.)

I would like to ask for some help with this. I was wondering what is the result of the commutator of the field for spacelike separated points but for $m^2 < 0$. To see this I tried to follow the same derivation than in Peskin & Schroeder, but I fail to see where the differences are, and thus I am forced to conclude that the commutator of the field at spacelike separated points does also vanish in case of $m^2 < 0$.

The Peskin & Schroeder derivation goes as follows. Assume a real scalar field that fulfils the Klein-Gordon equation:

$$\left( \frac{\partial^2}{\partial t^2} - \nabla^2 \right) \phi + m^2\phi = 0$$

Go to the momentum representation:

$$\phi(x) = \int \frac{dp^3}{\left(2\pi\right)^{3/2}}\, e^{ipx}\, \psi(p)$$

and get:

$$\left( \frac{\partial^2}{\partial t^2} - p^2 \right) \psi(p) + m^2 \psi(p) = 0$$

From this oscillator equation for $\psi(p)$, based on the assumption that the scalar field is real, defining creation and annihilation operators for the oscillator, and substituting above, one gets the following expression for the spatial part of the field:

$$\phi(x) = \int \frac{dp^3}{\left(2\pi\right)^{3/2}}\, \frac{1}{\sqrt{2E_{p}}} \left( a_p\, e^{ipx} + a_p^{\dagger}\, e^{-ipx}\right)$$

with $E^2_p = p^2 + m^2$. In the Heisenberg representation:

$$\phi(x) = \int \frac{dp^3}{\left(2\pi\right)^{3/2}}\, \frac{1}{\sqrt{2E_p}}} \left( a_p\, e^{iPx} + a_p^{\dagger}\, e^{-iPx}\right)$$

with $P$ the four momentum.

Then, the commutator of the field at spacelike separated points is:

$$\left[\phi}(x), \phi}(y) \right] = \int \frac{dp^3}{\left(2\pi\right)^{3/2}}\, \frac{1}{2E_p}} \, \left(e^{-iP(x-y)} - e^{iP(x-y)}\right)$$

with $(x-y)^2 < 0$

The integral can be separated in two ones

$$\left[\phi}(x), \phi}(y) \right] = D(x - y) - D(y - x)$$

and one of them can be rotated to get

$$\left[\phi}(x), \phi}(y) \right] = D(x - y) - D(x - y) = 0$$

Now, in case of $m^2 < 0$ the equation for the oscillator will lead to a repulsive oscillator but besides of this all the steps seem to be the same to me. Probably I am missing some obvious step, but I fail to see how $m^2 < 0$ changes something in this derivation.

 

[strangerep]

I would like to ask for some help with this. I was wondering what is the result of the commutator of the field for spacelike separated points but for $m^2 < 0$. To see this I tried to follow the same derivation than in Peskin & Schroeder,

Where specifically are you starting from in P&S? Most of the stuff in P&S leading up to (say)
their eqn (2.53) (ie your final eqn) already assumes real mass and +ve energy, so I'm not
quite sure how to relate their stuff to what you wrote.

but I fail to see where the differences are, and thus I am forced to conclude that the commutator of the field at spacelike separated points does also vanish in case of $m^2 < 0$.

The Peskin & Schroeder derivation goes as follows. Assume a real scalar field that fulfils the Klein-Gordon equation:

$$\left( \frac{\partial^2}{\partial t^2} - \nabla^2 \right) \phi + m^2\phi = 0$$

Go to the momentum representation:

$$\phi(x) = \int \frac{dp^3}{\left(2\pi\right)^{3/2}}\, e^{ipx}\, \psi(p)$$

This is not applicable here. It's only a 3D integral, but for a general treatment,
you'll need to use 4D Fourier transforms - see below.

and get:

$$\left( \frac{\partial^2}{\partial t^2} - p^2 \right) \psi(p) + m^2 \psi(p) = 0$$

You still have time derivatives there. With 4D transforms it would be $(E^2 - p^2)...$

From this oscillator equation for $\psi(p)$, based on the assumption that the scalar field is real, defining creation and annihilation operators for the oscillator, and substituting above, one gets the following expression for the spatial part of the field:

$$\phi(x) = \int \frac{dp^3}{\left(2\pi\right)^{3/2}}\, \frac{1}{\sqrt{2E_{p}}} \left( a_p\, e^{ipx} + a_p^{\dagger}\, e^{-ipx}\right)$$

with $E^2_p = p^2 + m^2$.

The above implicitly relies on P&S's eqn (2.40), i.e.,

$$\int \frac{d^3p}{(2\pi)^3}~\frac{1}{2E_p} ~=~\int \frac{d^4p}{(2\pi)^4}(2\pi)\, \delta^{(4)}(p^2 - m^2) \big|_{p^0>0}$$

As an exercise, try to derive this for yourself using contour integration in the complex
$p^0$ plane (for the standard real-mass case). Hint: Express the above delta fn
in the form $\delta^{(4)}\big((p^0 + X)(p^0 - X)\big)$ and use this formula:

$$\int dx\, \delta(f(x)) ~=~ \sum_k \frac{1}{\big|\frac{df}{dx}\big|_{x_k}} ~,$$

where the $x_k$ are the roots of the function $f(x)$ within the region
of integration.

After you've done this for the standard case, think about what changes
for imaginary mass, and without the constraint $p^0 > 0$.

[BTW, note that the measure in the integrals is written $d^4p$,
not $dp^4$ as in your post.]

 

[hellfire]

Thanks for your interest in this topic.

Where specifically are you starting from in P&S? Most of the stuff in P&S leading up to (say)
their eqn (2.53) (ie your final eqn) already assumes real mass and +ve energy, so I'm not
quite sure how to relate their stuff to what you wrote.

This is exactly what I would like to understand. Right now I do not understand where do they assume real mass, and what would change in the (2.53) result if the mass would be complex.

This is not applicable here. It's only a 3D integral, but for a general treatment,
you'll need to use 4D Fourier transforms - see below.

This I do not understand neither. The usual procedure, at least as known to me, is to start express the field as an expansion in eigenstates of momentum in a spatial hypersurface where it should be possible to impose the equal time commutation relations. In Minkowski space-time these are plane waves and thus one arrives to my expression above in the momentum representation. Afterwards, one plugs this expression into the equation of motion and gets a harmonic oscillator. Then, the harmonic oscillator is quantized as in NRQM imposing the commutation relations and the field can be rewritten as a function of the creation and annihilation operators. Up to this point the field operator is time independent. To get time dependence one changes to the Heisenberg picture and gets the expression I have written above. I do not understand why is it required to make use of 4D Fourier transforms. Actually, I do not understand neither why (2.40) of P&S is relevant here.

 

[strangerep]

Right now I do not understand where do [P&S] assume real
mass, and what would change in the (2.53) result if the mass would be complex.

OK, let's go back a bit further to P&S's eqn(2.20), i.e: the equal-time CCRs for a continuous
system. In particular, the 2nd part which says

$$[ \phi(\bold{x}) , \phi(\bold{y}) ] = 0 .$$

This is an assumption, motivated by imagining infinitesimal harmonic oscillators at
every point of spacetime. This is where the restriction to non-tachyonic fields is silently
introduced. But that choice of CCR was itself motivated by considering harmonic
oscillators, so the assumptions therein must be looked at more closely.

Also, look a bit further down the page at eqns (2.22) and (2.23), i.e.,

$$\omega_{\bold{p}} = \sqrt{|\bold{p}|^2 + m^2} ~~~~~ (2.22)$$

If m is imaginary, $\omega_{\bold{p}}$ is no longer real in general.
Similarly, looking at:

$$\phi = \frac{1}{\sqrt{2\omega}}(a + a^\dagger) ~~~~~~ (2.23)$$

we see that $\phi$ is no longer guaranteed to be Hermitian, if $\omega$ is complex.
That makes it extremely difficult to get a sensible Hilbert space with +ve-definite
Hermitian inner product.

Basically, all the usual Fourier expansions assume that one can decompose
the field into (sums/integrals of) sines and cosines. But if the harmonic oscillator
eqn has imaginary mass, one must use sinh and cosh instead, and vast amounts
of the familiar Fourier machinery from the real-mass case become inapplicable.

I do not understand why is it required to make use of 4D Fourier transforms.
Actually, I do not understand neither why (2.40) of P&S is relevant here.

I suggested you look at 4D Fourier transforms to get an idea of what goes wrong.
Proving (2.40) gives insight into the way the $p^0$ poles occur on the real axis, and
by restricting to positive energy, we choose whether to close the contour in the upper
half place, and how to deform it around the poles to conform with our choice.
For imaginary mass, the poles are no longer on the real axis, and it is no longer
physically reasonable to restrict to one sign of $p^0$ (because here it's possible
to pass from +ve to -ve $p^0$ via a continuous Lorentz transformation).

But maybe the essence of your original question is more directly addressed by
just looking a bit further back, as I've sketched above.

 

[hellfire]

Ok, this clarifies it a bit more. I understand your points and I will take a closer look to that. I realize however that I cannot follow the same derivation than for m² > 0 to get some conclusion, but, however, I still do not know if the spacelike commutators for m² < 0 vanish.

I have, by the way, found this:
http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/tachyons.html
why does not point 1) (at the end) imply that classically tachyonic solutions should not spread faster than c and thus the spacelike commutators should vanish?

 

[strangerep]

Ok, this clarifies it a bit more. I understand your points and I will take a closer look to that. I realize however that I cannot follow the same derivation than for
m² > 0 to get some conclusion, but, however, I still do not know if the spacelike commutators
for m² < 0 vanish.

I pretty sure that spacelike commutators for a tachyonic field should not vanish in general.
But I'm no longer sure what you've really trying to achieve. The choice of commutation
relations is an arbitrary decision during the procedure of quantization - to be compatible
with a particular classical model.

I have, by the way, found this:
http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/tachyons.html
why does not point 1) (at the end) imply that classically tachyonic solutions should not spread faster than c and thus the spacelike commutators should vanish?

But those solutions look like waves whose amplitude increases exponentially as
time passes, (and was presumably infinitesimal in the infinite past).
Physically sensible normalization might be difficult. :-)

Also, how do you pass from "localized tachyon disturbance does not spread faster than c"
to "spacelike commutators should vanish"? What about the spacelike commutators inside
the localized region? Anyway, those solutions have other nonsensical properties if you
think about t<0.

 

[Haelfix]

Tachyon commutators can and usually do vanish outside of the lightcone in field theory at least for physically realistic theories.

The prototypical example is the Higgs mechanism, where you have a tachyon at the top of the potential and the tachyons then undergo condensation. Ultimately it arises b/c of nontrivial vacuum configurations usually associated with spontaneous symmetry breaking.

Now, in the case where you have a true tachyon in the theory, I suspect the problem is deeper in that you will find processes that violate unitarity.