The first couple of pages are filled with a content-free bitterness about the path integrals and a unsubstantiated promotion of algebraic quantum field theory: the kind of silly unphysical whining that all of us know very well from "Not Even Wrong" and other places on the Internet. The author is upset about the "string theory caravan" that does not support "great" ideas - such as the "great" idea of Prof. Schroer himself that the path integrals are bad.
The first non-trivial statement appears on page 3. Prof. Schroer essentially claims that the path integrals give a wrong result if you use them to describe a spinning top. The critical sentence is the following:
- The paradoxical situation consists in the fact that although the higher fluctuation terms (higher perturbations) are nonvanishing, one must ignore them in order to arrive at the rigorous result.
Spinning top
So what about the spinning top in quantum mechanics that Prof. Schroer believes to invalidate the path integral at higher orders? A spinning top is described by
- the center-of-mass position
- the orientation
In the wavefunction approach to quantum mechanics, the wavefunction is therefore defined on the SO(3) group manifold. Quantum mechanics on curved manifolds can also be interpreted as a special (one-dimensional) example of non-linear sigma models, a kind of theory that we are solving all the time (mostly using path integrals).
This SO(3) group manifold is essentially the three-sphere (divided by a Z_2 symmetry). The kinetic energy is a second-order differential operator on this three-sphere and its eigenvalues are given by simple expressions of the form "j(j+1) + j'(j'+1)" if I did it correctly. That's the spectrum of the Hamiltonian, up to trivial factors.
Path integral for spinning tops
Can one obtain it from the path integral? The answer is, of course, a resounding Yes. Even Schroer admits that by expanding the action around the stationary points - which happen to be saddles in this case, one obtains the correct result in the semiclassical approximation. However, he argues that there are higher-order terms that are nonzero and that would give us a wrong result if we included them.
Of course that he is wrong. These terms cancel. Open, for example, this paper by
At the end of page 7, they explain that the higher-order terms cancel. In fact, on page 8, they also tell you a simpler way to see why they do: just use the coordinates for the perturbations in which the action is exactly bilinear. You can always find such coordinates in which the vanishing of the higher-order terms becomes trivial. If you get any other result with different coordinates, then it means that you have modified the short-distance physics on the three-sphere.
Your new physics differs from the regular local physics in flat space; in the wavefunction approach, such a modification would correspond to an addition of higher-derivative operators into your Hamiltonian: a typical UV modification. You can view such UV modifications as higher-order operators or as generalizations of the ordering ambiguities in quantum mechanics.
The fact that the correct amplitudes are obtained to all orders is a matter of common sense. It has to work. The path integral knows about all the symmetries and of course it preserves them whenever it can. It also informs us whenever the symmetries can't be preserved - for example, the anomalies are represented by linear divergences.
The muscles of the path integral in quantum mechanics
There are some cute examples in quantum mechanics where the path integral can show its muscles and lead us to exact results. For example, we know from the Hamiltonian approach that the Hydrogen atom conserves the Runge-Lenz vector "A_i" which means that the SO(3) symmetry is actually enhanced to SO(4): "A_i" are interpreted "J_{i4}" times some function of the Hamiltonian (the ordering does not matter because the Hamiltonian commutes with all the SO(4) generators).
This means that you can find the Hydrogen atom discrete spectrum without solving any differential equations whatsoever - just by group theory. Its SO(4) symmetry is isomorphic to "SU(2) x SU(2)" except that you can prove that the second Casimir invariants of these two SU(2) factors must be equal - their difference is proportional to the inner product of the angular momentum and the Runge-Lenz vector which vanishes because they are orthogonal to each other.
This means that under the "SU(2) x SU(2)" group, the eigenstates of energy transform as "(n,n)" where "n=2j+1" is the degeneracy of the representation of SU(2) with spin "j". You can see that the total degeneracy of this energy level is therefore "n^2", as taught in high school chemistry classes, and because of the "function of the Hamiltonian" mentioned previously - a power law that you can derive by computing the commutators of the Runge-Lenz components - one can also prove that the energy of the states transforming as "(n,n)" equals "-E_0 / n^2" as it should. For positive energies, the group is SO(3,1) and a continuum spectrum is possible.
Hydrogen atom path integral
That's very nice - the Hydrogen atom is solvable. Is it also solvable using the path integrals? The answer is, of course, yes. As Kleinert mentions in his 2004 book, Feynman himself challenged him:
- “Kleinert, you figured out all that group-theoretic stuff of the Hydrogen atom, why don't you solve the path integral!"
They must insert the correct delta-function of "t_{final} - t_{initial} - integral" where the integral is calculated from their auxiliary time. The delta function is expressed using the standard Fourier-transformed formula as an extra integral. They add another dummy path integral over "x_4(t)" and "p_4(t)" and eventually convert the whole calculation to the path integral for the four-dimensional harmonic oscillator. Nice and elegant. Exact.
Message
The message is that every known well-defined problem in any quantum mechanical system with a classical limit that is solvable using other methods is also solvable using path integrals. And of course, one obtains the right results. Moreover, path integrals are superior in their treatment of non-perturbative corrections (instantons) and gauge symmetries (Faddeev-Popov ghosts). They have become the canonical way to define the Lorentz-covariant prescription for perturbative amplitudes in string theory. Attempts to remove path integrals from physics in 2006 are plain ludicrous.
There can exist systems whose full physics cannot be defined as a quantization of a classical system - and indeed, the generic points in the string/M-theory moduli space are believed to be an example. In these cases, any approach based on a classical starting point - including the path integral approach - is expected to be incomplete. But one needs new principles and restrictions that define such theories without a "canonical" classical limit. A content-free and unjustified criticism of the path integral approach is certainly not good enough to make any progress.
