Divergences in Second Quantised Field Theories

Divergences in the Field-In-A-Box Model

In second quantised field theories, there are two “divergences to infinity” that one might find offputting (at least I did at first) to do with the quantum ground state. One thinks of the universe as a “box” and then the (nonrelativistically) second quantised field is essentially a collection of nonrelativistic quantum harmonic oscillators, one for each of the box’s modes with wavevector $(k_x, k_y, k_z)$. As you think of the box getting bigger and bigger, the packing density in momentum ($\mathbf{k}$)-space gets denser and denser.

Now the quantum ground state of each oscillator of frequency $\omega$ has energy $\hbar \omega / 2$ so that it fulfills the Heisenberg uncertainty principle. Therefore, the total ground state energy for all the modes with frequency less than some “cutoff” frequency $\omega_0$ in a box of sidelengths $(L_x, L_y, L_z)$ is:

$$\sum\limits_{4\pi^2\,c^2\left(\frac{n_x^2}{L_x^2}+\frac{n_y^2}{L_y^2}+\frac{n_z^2}{L_z^2}\right) <\omega_0^2} \hbar \;c\;\sqrt{\frac{n_x^2}{L_x^2}+\frac{n_y^2}{L_y^2}+\frac{n_z^2}{L_z^2}}$$

an integral which of course diverges as the mode spacing in momentum space approaches nought and we replace the ground state energy sum with an integral, even if we limit the frequencies to ones below a cutoff $\omega_0$! Oh dear: Planck makes his postulate to clear up an ultraviolet catastrophe in Rayleigh Jeans theory only to lead ultimately to another divergence.

But this particular divergence is altogether physically reasonable, because the divergent energy is in proportion to the box volume. The energy “cost” per unit volume to set up a quantum electromagnetic field is a perfectly well defined, finite number. So now imagine being some god (as one does when off one’s chlorpromazine medication) who wants to create a universe: so you log into, say Odin’s account, at “buildmyuniverse.com”. As you’re dabbling with creation, the website will tell you that you need to pay an amount of energy to establish a quantum electromagnetic field in proportion to the volume of the universe you want to build!

Less flippantly, modern physics doesn’t believe empty space is empty: space is the quantum fields that fill it. If the universe is big, then the packing density of its momentum space is high and therefore so is the energy needed to establish it.

The other divergence one comes across is the ultraviolet divergence one lets $\omega_0\to\infty$. This is handled by “renormalisation” procedures, which are a way of asserting that we dont really know what happens at very short length scales, so we set up our calculations so that other known physical data can implicitly set the integral value as $\omega_0\to\infty$. It is an assertion of our ignorance so far, and means that other physical data (such as electron self energies) must be put into the theory by hand rather than naturally being derived from it. Renormalisation is like leaving a software “hook” in a theory: it arranges asymptotic behaviour so that the behaviour we foretell from our physics is independent (within reasonable limits) of the kind of cutoff that some future experiment result might show us how to put into our theories.

Separability of the Quantum Hilbert State Space

There is another factor we to consider here and that is the Wightman separability axiom: that we only deal in quantum theories with a separable Hilbert space. In second-quantised theories, this axiom has a very clear and practical physical  meaning – “we must only put a finite number of particles into a finite quantisation volume”.

We take a Fermionic Fock space for a finite quantisation volume (“Box”), so that there are countably infinitely many plane wave as the “modes”. Then an arbitrary basis member is of the form:

$$\left|\left.0,1,1,0,1,1,1,0,1,0,\cdots\right>\right.$$

that is, a countably infinitely long string of 0s and 1s showing which modes are filled. This set is mapped bijectively to the interval $[0,1]$ – it’s simply a binary expansion of a number in $[0,1]$. So boom! The old $\aleph_1$-in-the-Fock trick: we’ve got ourselves a Hilbert space with $\aleph_1$ basis states just like that! (call it $\beth_1 = 2^{\aleph_0}$ if you want to forswear the continuum hypothesis!)

The problem “only gets worse” for Bosonic Fock spaces of course (although we’re still “only” dealing with $\aleph_1$).

One can tame this space by saying we consider the space of states with a *finite* number of particles in them. Then our basis Fermionic Fock states are equivalent to finite binary expansions of numbers in $[0,1]$, i.e. to a proper subset (those with nonrecurring finite binary expansions) of the rationals $\mathbb{Q}\cap[0,1]$ in that interval. So now we have our friendly separable Hilbert space with a countably infinite basis back again.

How needful is the separability axiom? Clearly it makes life easier and is mathematically reasonable. This example shows that there are readily conceivable quantum spaces (although one might argue how “physical” an infinite number of particles in a finite quantisation volume is) that have distinctly different basis cardinality. So whether or not you choose to take it on does make a difference to your physics. The example also shows how in this case you can make a physically reasonable argument that our real quantum space is spanned by a meagre subset of the uncountable basis states. It’s worth saying again to emphasise that that separability axiom means real, practical physics here: – “only a finite number of particles for a finite quantisation volume”.