Given a higher derivative classical/quantum field theory with say one scalar field, particularly the Lee-Wick standard model. It has been shown that such a field theory encompasses two kinds of fields, one normal and other a ghost field, in a particular limit, generally the high mass limit of the ghost field. We started with a single field but in ended up with a two particle field theory.

This makes me ask:
Is there any general mathematical theorem by which we can show that a $n$th order derivative theory(scalar, spinor or vector) can be quantized into $n$ different kind of particles? Has it got to do something with the symmtries involved in the theory?
And what is the physical importance of the constraint? And how exactly is the system behaving when that constraint is not satisfied?(It seems to be in some sort of a enmeshed state of the two particles.)

My question mainly relies on the paper in the following link [The Lee-Wick standard model]: http://arxiv.org/abs/0704.1845 . In section 2 (A Toy Model) he considers a higher derivative Lagrangian eq.(1) for a scalar field $\phi$. And then introduces an auxiliary field $\tilde{\phi}$. And then it is shown that given the condition $M > 2m$, the Lagrangian sort of gets decoupled into two different fields. One normal field and other a ghost field, both interacting with each other.


Most higher derivative theories —and in particular Lee-Wick's model— do not have ghost excitations but they are unstable (Hamiltonian unbounded from bellow). Yes, almost everyone says the opposite but all them are unfortunately wrong. They do not quantize the theory properly. Whenever a degree of freedom has negative energy at the classical level, it must have negative energy at the quantum level as well. People try to fix the problem of negative energy exchanging the frequencies between creation and annihilation operators at the price of introducing ghosts. But this is not the right way of quantizing the theory because the classical limit is totally wrong. One may canonically quantize the harmonic oscillator with an additional higher derivative term to convince himself of what I am emphatically claiming.

Anyway, either one quantizes the theory correctly getting an unstable theory or one quantizes the theory incorrectly getting a theory with negative norm states, the quantum theory does not make any sense. It is not a quantum theory.

Is there any general mathematical theorem by which we can show that a nth order derivative theory can be quantized into n different kind of particles?

The classical and the quantum theory must have the same number of degrees of freedom (DOF). The classical theory has half DOF of the number of initial conditions must be given to determine a solution. In a normal field theory —let's say Klein-Gordon— one must specify initial value of the field and momentum (or velocity) for every field. Thus one has one DOF for a real field, two for a complex field, four for two complex fields. When one adds higher time derivatives, one requires more initial conditions to know a solution (initial accelerations, initial fourth derivatives, etc). When there are constrains the counting of degrees of freedoms is a little bit more subtle. For instance, electrodynamics is a second-order theory and the electromagnetic four-potential $A_{\mu}$ has four components so one could naively think that the theory has four degrees of freedom, but this is not true because there are gauge redundancies (there are first class constraints) that make different (gauge related) $A_{\mu}$ correspond to the physical situation. So that the theory has only two DOF corresponding to the two polarizations of electromagnetic waves. Regarding the quantum theory, the number of degrees of freedom corresponds to the number of particles (each physical polarization counts as a particle).

  • $\begingroup$ Regarding ur first comment Are you saying higher derivative QFT's are inconsistent and should be thrown away? $\endgroup$ – Omkar Sep 1 '12 at 5:17
  • $\begingroup$ Regarding your second comment, I can seem to agree with it in normal derivative QFT, but as you can see in HD QFT it can be difficult in general to find the no. of particles, as they seem to be decoupled only in a particular limit. Hence I was thinking of an alternative method, something like a relation between symmetry and no. of particles? $\endgroup$ – Omkar Sep 1 '12 at 5:30
  • $\begingroup$ @Omkar 1) Not all, but most of the HDT existing in the literature are inconsistent. See "Ostrogradski instability". 2) In a theory with third order time derivatives, you need to specify 4 initial data, therefore there are 2 physical DOF. I don't know what is confusing you (?). $\endgroup$ – Diego Mazón Sep 1 '12 at 19:40
  • $\begingroup$ Oh, I just realize what is probably confusing you. The condition $M>2m$, right? If this did not verify, then the Hamiltonian would be complex, there would be dissipative terms and the Lagrangian in Eq. 8 would be complex. Therefore the total probability would not be conserved in time. $\endgroup$ – Diego Mazón Sep 1 '12 at 21:53
  • $\begingroup$ 1) Do u think the inconsistency on those theories can be removed if we quantize them correctly(or diffrently)? Does it make sense to ask such a question? 2)By DOF I mean the different kinds of particles the theory quantizes into. And I cant see why in HD QFT it should depend on a condition of some free parameters. Also what you are saying makes total sense for scalar normal derivative theories, but say u have a dirac field, first order spinor field and it quantizes into 4 different particles. Hence I am unable to see the pattern, like what would it be for 2nd order spinor fields? $\endgroup$ – Omkar Sep 2 '12 at 7:16

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