Re: Questions about Higgs scalars

Jay R. Yablon wrote:

[...snip questions 1)-5)...]

These were all questions that tried to relate the single particle wave
function to a dynamical scalar field in field theory.

6) Am I really asking how one goes from treating a Higgs scalar field to
treating a single Higgs particle? Is there a better way to ask all of
this?

This question actually addresses the crux of the matter. And it is the
one that I will address.

First, let me come right out and say it: dynamical scalar fields are
not wave functions, nor vice versa. And since they are not the same
thing, it scarcely makes sense to compare them; apples and oranges,
you see. Even though, wave functions and scalar fiends sometimes happen
to satisfy the same equation. However, that's mostly because there
aren't that many linear second order partial differential equations
that are invariant under the appropriate symmetry group (Galilei,
Poincare, or even Euclidean).

There is a relation between them however. Unfortunately, this relation
is rarely addressed in field theory text books that come from a
high-energy background. One prominent exception is Weinberg's QFT
vol.1. It is addressed in some older books like Dirac's Principles of
QM and some old fashioned books on many-particle physics, the latter
coming from a condensed-matter background.

This relation I've dubbed, for lack of a better name, the Fundamental
Theorem of Second Quantization. It can be stated roughly as follows:
the quantum theory of a variable number of identical particles is
equavalent to the quantum theory of a field.

Equivalence requires two arrows. The arrow from the many-particle
theory to field theory is called second quantization and is described
in many books. What many books do not explicitly mention is that there
is a reverse arrow going from field theory to many particle theory. Let
me sketch how identification is done in both directions.

Say you have a quantum theory of a single particle. It's Hilbert space
is given as the space of complex square-integrable functions of spatial
coordinates. Observables, like position and momentum, are represented
by multiplicative or differential operators acting on this space. If
you want to deal with two particles, you must take instead functions of
two copies of spatial coordinates. You must also make sure that they
are only either symmetric under the interchange of particle coordinates
or antisymmetric (this implements the identical particle requirement).
And so on, for a larger number of particles. If you wish to consider a
variable number of particles, then you must enlarge the Hilbert space
of the theory to be the sum of all multi-particle Hilbert spaces, plus
a one-dimensional subspace corresponding to the presence 0-particles.
Then, you can introduce creation and annihilation operators that,
respectively, add or destroy a particle in a particular state. These
ladder operators may be index by any orthonormal basis in the
single-particle Hilbert space, or alternatively by wave number or
position. When indexed by position, they can be written as Phi(x),
where I've used a capital letter to indicate that we have an operator
for each value of x. The important part is that the (anti-)commutation
relations satisfied by these operators are precisely the same as the
canonical (anti-)commutation relations that must be satisfied by the
quantized observables representing the values at different spatial
points of a scalar field phi(x), where the lower case phi now
represents a classical field. In other words as we were constructing
the quantum theory of a variable number of identical particles, we've
inadvertantly constructed the quantum theory of a classical field.

Now, what do we do in the reverse direction? Suppose we are given a
quantum field theory. Its Hilbert space of states is the Fock space
written as a direct sum of subspaces with different numbers of field
quanta, the vacuum state |0> representing zero quanta. Its observables
are operators on this space. For example, the field operators Phi(x)
represent the quantization of the values the field takes a different
spatial points. Let |psi> be a state with precisely 2 quanta. Consider
the expectation values of the form <0|Phi(x)Phi(y)Phi(z)...|psi>. By
the virtue of the number of quanta it contains only the products
psi(x,y) = <0|Phi(x)Phi(y)|psi> are non-vanishing; similarly for
psi*(x,y) = <psi|Phi(y)Phi(x)|0>. The canonical (anti-)commutation
relations of the field operators make sure that the function psi(x,y)
is (anti-)symmetric in its arguments. Also, inserting a resolution of
identity into the inner product <psi|psi> and doing some integrals
ensures that psi(x,y) is square integrable. Functions psi(x,y,z,...)
with similar properties but a different number of arguments can be
constructed starting with a Fock state containing a different number of
quanta. But these are precisely what were called multi-particle wave
functions in the previous paragraph. You can now obtain the single
particle wave functions by restricting your attention the subspace of
single-quanta Fock states through the relation psi(x) = <0|Phi(x)|psi>.
There is also a procedure for translating operators from Fock space
operators to operators acting on wave functions.

The above discussion needs no modification only for the case of
non-relativistic paritcles and non-relativistic fields. In the
relativistic case, the wave function formulation of particle mechanics
runs into infamous interpretational issues. Namely, the naive
probabilistic interpretation of wave functions no longer applies.
However, that is no great loss. We have a satisfactory constructions of
relativistic quantum field theories. Then, the single particle version
of the corresponding particle theory is obtained from the single-quanta
Fock states. The probabilistic interpretation is the repaired by
relying on the existing Fock space inner prduct, which through the
above equavalence can be expressed in the language of wave functions.
Thus, the normalization of a wave function derives solely from the
normalization of the Fock state to which it corresponds.

Finally, the question: which PDE does the wave function satisfy? The
field operators, Psi(x,t), can be written in Heiseberg picture so that
they are functions of both spatial coordinates and time. If the field
theory is non-interacting, then the field operators will satisfy the a
PDE derived from its Lagrangain (usually, the Schroedinger, Dirac,
Klein-Gordon, or other equations). In the Heisenberg picture, the
states carry no dependence on time, so the wave function
psi(x,t)=<0|Psi(x,t)|psi> will satisfy the same PDE. However, when the
field theory is interacting, the field operators satisfy a non-linear
operator equation. Translating this statement into the language of wave
functions means that the multi-particle will in general satisfy an
infinite dimensional system of coupled PDEs (which allow for the
creation and destruction of particles). Unless, that is, the number of
field quanta is a conserved quantity. Then the PDEs for wave functions
of different numbers of particles decouple (this situation is usually
seen in the non-relativistic case).

The lesson to be drawn from all this is that wave functions and quantum
fields are not the same. However, they can be related to each other in
a precise manner. If you want to see a more technical discussion of
this connection, I suggest the following references:

P.A.M Dirac, Principles of Quantum Mechanics, Secs. 59 and 65.
S. Weinberg, The QUantum Theory of Fields I, Chap. 14.

Any other book that deals with second quantizaton, if you are not
familiar with it, will also be helpful.

Hope this helps.

Igor

.

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