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Quantum fields are presented as operator-valued distributions, so that the operators in the theory are linear functionals of some test function space. This works well for free fields, giving us a particular form for VEVs, in which the 2-point connected correlation function determines the whole theory, and a trivial S-matrix.

For interacting fields, however, the VEVs and the statistics of S-matrix observables that are predicted by the theory depend not only on the test functions (generally given for the S-matrix in terms of pure, improper wave-number modes of the free field at $t=\pm\infty$) but also on the "energy scale of measurements". One obtains a different functional form at different energy scales.

The energy scales at which one is measuring are already encoded in the wave-numbers associated with the test functions, so it seems that the test functions play extra duty, to determine the functional form of the VEVs (at least in their S-matrix form) both by linearly smearing the quantum field and, more indirectly, by determining the renormalization scale. As a consequence, the VEVs, the Wightman functions, seem clearly to be nonlinear functionals of the test functions for interacting QFTs.

I've worried at this for a while. The argument as put brings into question the Wightman axioms' insistence that a quantum field must be a linear map from the test function space to the space of operators, although I suppose the effect would be only logarithmic in some measure of the frequencies of the test functions. The Haag-Kastler additivity axiom also seems problematic. I'm not aware of weakening this particular aspect of the Wightman or Haag-Kastler axioms so as to be able to construct models of the axioms that parallel empirically successful QFT having been discussed in the literature, but have they?

[Note that dropping the linearity of the map from test functions to operators in the algebra of observables does not affect the linearity of the algebra. There's still a perfectly good probability interpretation of a state over a *-algebra and a perfectly good Hilbert space, but the relationship of measurements to space-time is somewhat modified.]

EDIT: I'll attempt to make the Question slightly more accessible. It turns out that what follows makes it significantly longer. The Wightman functions (effectively a different name for Vacuum Expectation Values (VEVs)) allow the reconstruction of a Wightman (quantum) field. That is, given the vacuum state over the algebra one can construct the Wightman functions, $$W(x_1,x_2,...x_n)=\left<0\right|\hat\phi(x_1)\hat\phi(x_2)...\hat\phi(x_n)\left|0\right>,$$ but also, given the Wightman functions one can construct the algebra. Wightman functions satisfy somewhat arcane relationships, so that just any function $VV(x_1,x_2,...,x_n)$ is almost certainly not a Wightman function and cannot be used to reconstruct a Wightman field.

The object $\hat\phi(x)$ is not an operator in the algebra of observables, it is an operator-valued distribution, so that if we have a well-behaved "test function" $f(x)$, where "well-behaved" most often means that it is smooth (infinitely differentiable) and has a smooth fourier transform, then $\hat\phi_f=\int\hat\phi(x)f(x)\mathrm{d}^4x$ is an operator. This is often called "smearing" with the test function $f(x)$. The test function $f(x)$ effectively tells us what relative amplification we apply to the field at each point in space-time for a given measurement, and its fourier transform $\tilde f(k)$ tells us what relative amplification we apply to the field at each point in momentum space. In signal analysis the same object is called a "window function", which can be looked up on Wikipedia. One can talk about the bandwidth of a measurement, or say that a signal transformation is high-pass or low-pass. If it were not for non-commutativity of measurement operators at time-like separation, quantum field theory would be exactly what one would want for a mathematical formalism for modeling stochastic signal processing. If one only talks about measurements that are space-like separated —admittedly a tight constraint—, QFT is a stochastic signal processing formalism.

In contrast to $\hat\phi(x)$, $\hat\phi_f$ is not a singular mathematical object; the difference is comparable to the difference between a smooth function and the Dirac delta function. In terms of these operators, the Wightman functions become $$W(f_1,f_2,...,f_n)=\left<0\right|\hat\phi_{f_1}\hat\phi_{f_2}...\hat\phi_{f_n}\left|0\right>,$$ which is linear in all the test functions. When we move to the practical formalisms of interacting QFT, however, the VEVs predicted by the bare theory are infinite. Renormalization fixes that in a Lorentz invariant way, but it leaves the VEVs a function of the energy scale $\mu$ of the measurements that are involved in an experiment. Thus, one finds that the Wightman functions are now a function of $\mu$ as well as of the points $x_i$, $W_\mu(x_1,x_2,...x_n)$, or, in terms of operators, $W_\mu(f_1,f_2,...f_n)$. [It's perhaps worth noting that we don't know whether these $W_\mu(...)$ satisfy the relationships required for them to be Wightman functions, so that we could reconstruct a Wightman field, but I've never seen a discussion of perturbative QFT conducted in such terms.] The Wightman functions in terms of positions do not encode energy scales, but they are not observables, they are only a template for constructing observables. The fourier transforms of the test functions, $\tilde f_i(k)$, however, already determine various energy scales at which the measurements made in an experiment operate, potentially in much more detail than a single number $\mu$. We could just say that the energy scale of measurements $\mu$ is separate from the energy scales determined by the test functions $f_i$, but it seems more natural to me to say that $\mu$ is a functional of the test functions $f_i$, in which case we can write $W'(f_1,f_2,...f_n)=W_\mu(f_1,f_2,...f_n)$, where $W'$ is now a non-linear functional of the $f_i$. I suppose we expect $W'$ to be a symmetric function of each of the $f_i$.

For a more-or-less trivial example, suppose we have two ordinary quantized Klein-Gordon fields $\hat\phi_f$ and $\hat\xi_f$, not necessarily having the same mass spectrum, then we could construct a quantum field $\hat\Phi_f=\hat\phi_f+\hat\xi_{f+\epsilon f^2}$ (where the square $f^2$ is $[f^2](x)=[f(x)]^2$, so that $\hat\Phi_f$ satisfies microcausality), so that $\hat\Phi:\mathcal{S}\rightarrow\mathcal{A};\hat\Phi:f\mapsto\hat\Phi_f$ is a nonlinear map from the test function space into the algebra of observables. This is still a Gaussian field, so not interacting. Already there start to be significant complications. One of the most fundamental Wightman axioms is the restriction to positive spectrum, which in terms of test functions is a projection to forward light-cone components of the test functions. Only components of $\tilde f(k)$ for which $k_\alpha$ is in the forward light-cone have any effect on measurement results. In the above, however, forward light-cone components of $f^2(x)$ contribute, which in fourier space, where $f^2(x)$ is a convolution, means that negative frequency components of $f(x)$ will contribute. Although this means that this kind of nonlinearity apparently does not satisfy positivity of the energy, it's not clear that this means such a system is not stable, which is the (non-axiomatic) reason why we insist on positivity of the energy as an axiom. Which, whoops, brings into question positivity of the energy, one of the big axioms, which I asked about here.

Now, of course, I'm trying out doing research here as if it was a virtual blackboard. Many of the ideas above ought to be rubbed out, different ideas tried and equally or more vigorously discarded. I've got lots of pieces of paper that have discarded ideas on them that you don't want to see, because I hope you have your own discarded ideas, as anyone who doesn't get too fixed in their ways should, most of which I probably don't want to see (not that I'm not a little fixed in my ways, but I've discarded some ideas). The trouble is that I can't see how we can go forward from me just typing away, I think the feedback loop may be too long for this to work. Still, the alternative is to write a paper and get feedback six months later. This introduces Questions that I may try to put on meta.

By the way, making a Question Favorite counts for no Reps, but it's much appreciated.

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I think I just barely understand this question, but it seems like a great one and I hope someone answers it properly. – Keenan Pepper Apr 26 '11 at 6:49

3 Answers 3

up vote 4 down vote accepted

Field theories are nonlinear because the quantum fields satisfy nonlinear dynamical equations.

But renormalization does not make quantum fields into a nonlinear functional of test functions. The Wightman distributions are, by definition, linear functionals of the test functions, and Wightman distributions always encode renormalized fields.)

Instead it changes the space of test functions to one where the interacting quantum fields are perturbatively well-defined. This gives a family of representations of the field algebra depending on an energy scale. All these representations are equivalent, due to the renormalization group, and the corresponding Wightman functions are independent of the renormalization energy. (In simpler, exactly solvable toy examples that need infinite renormalization, this can actually be checked.)

The dependence on the energy scale would not be present if contributions to all ordered were summed up (though nobody has the slightest idea how to do this nonperturbative step). The energy scale is simply a redundant parameter the influences the approximations calculated by perturbation theory.

The renormalization group is an exact but unobservable symmetry (just like gauge symmetry) that removes this extra freedom, but as computations in a fixed gauge may spoil gauge-independence numerically, so computations at a fixed energy scale spoil renormalization group invariance numerically.

Note that Wightman functions are in principle observable. Indeed, the Kadanoff-Baym equations, the equations modeling high energy heavy ion collision experiments. are dynamical equations for the 2-particle Wightman functions and their ordered analoga.

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Thanks, Arnold. There are so many different ways to say this stuff. I've never seen "it changes the space of test functions" before. Food for thought. – Peter Morgan Mar 19 '12 at 19:23
@Peter: This is my way to express Haag's theorem. In the cutoff theory, you do a unitary transformation from the bare to the renormalized basis. In the limit, this transformation becomes ill-defined, as the limiting representations are inequivalent. Read the chapter on renormalization in my FAQ for details. – Arnold Neumaier Mar 19 '12 at 20:04

G. Scharf, instead of renormalizations, splits arguments in the distributions. I think it is equivalent to replacing one of distributions with a regular function.

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I thought I might hear from you! Your Answer is short enough that I feel that I have to read between the lines a little to give you the +1, but hey! Your summary of the Scharf seems a little cryptic, but it also seems a little evocative and relevant to the way I've asked my Question (having said it's Useful, I have to add that I don't feel I can call it an Answer). I think I've looked at Scharf in the past, but before my current attempts to get a handle on interacting quantum fields, so a new look would be good. Have to get hold of it from the library and look again. Thanks. – Peter Morgan Apr 26 '11 at 19:40

This paper by Borcherds offers a dense treatment of perturbative renormalization in an altered version of Wightman's axioms. There is no failure of linearity in the test functions anywhere - see especially the bottom of page 8.

I believe the problem in your reasoning is your main assumption that test functions define absolute energy scales. At least in the context of Borcherds's treatment, Feynman measures are essentially made from the data of energy cutoffs (which form an infinite dimensional space of possibilities), and the Wightman distributions are constructed from a choice of a Lagrangian and a Feynman measure, so the energy scale is an input to the distribution. One may then probe the distributions with test functions, but these test functions are not connected to the energy scale that you chose when constructing the distribution. In fact, there is an infinite dimensional ultraviolet group (also called the group of renormalizations) that acts both on the space of Lagrangians and on the space of Feynman measures, such that the resulting Wightman distributions are fixed. This group does not act on the space of test functions.

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Definitely +1, albeit after reading only 3 pages. I'm very surprised I didn't leap at this paper when it appeared on arXiv, but I missed it. I love Borcherds' reconstruction of QFT here! Wonderful to see someone stepping out of the algebraic tradition box. More response from me later, I have little doubt. Concerning your 2nd para, I presented test functions as defining energy scales associated with a specific set of measurements, not as defining absolute energy scales. But when I've read more Borcherds, I hope I'll better understand your Answer. – Peter Morgan Jun 28 '11 at 14:42

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