Physical Meaning of a Quantum Field

[Bobjoesmith]

TL;DR Summary

I am wondering how a quantum field can be represented as a tensor product of a vector space and an endomorphism of a subspace of a Hilbert space (which is how it is represented in a paper I am reading about the Whitman Axioms), and what this tensor product actually represents.

Sorry in advance if this question doesn't make sense.
Anyway, I am reading a paper about quantum field theory and the Whitman Axioms (http://users.ox.ac.uk/~mert2060/GeomQuant/Wightman-Axioms.pdf), and it describes a field (Ψ) as

Ψ:𝑀→𝑉⊗End(𝐷)

where 𝑀 is a spacetime manifold, 𝑉 is a vector space, and 𝐷 is a dense subspace of a Hilbert space. My question is what 𝑉⊗End(𝐷) physically represents?
Once again thanks for any help.

 

[DarMM]

First of all the notes are slightly incorrect with the name. Those are the Wightman-Gårding axioms. The Wightman axioms are a second set of axioms for the expectation values of fields. The reconstruction theorem of Wightman, Schmidt & Baumann shows these axioms are equivalent.

(There are two other sets of axioms in QFT: Osterwlader-Schrader and Fröhlich)

In what you have given there the field is defined as a map:
$$\Psi : M \rightarrow V\otimes End\left(\mathcal{D}\right)$$
So $End\left(\mathcal{D}\right)$ represents that the fields won't be defined on the whole Hilbert space, but only a dense subset of it $\mathcal{D}$. That they be not just any set of maps on $\mathcal{D}$ but endomorphisms is crucially important in order to define expectation values, because if the fields mapped out of this space you wouldn't even be able to define a (smeared) two-point function:
$$
\left(\Omega , \phi\left(f\right)\phi\left(g\right)\Omega\right)
$$
With $\Omega$ the vacuum. If $\phi\left(g\right)$ acted on the vacuum to a vector outside the domain of the (smeared) fields then the action of $\phi\left(f\right)$ on $\phi\left(g\right)\Omega$ would be undefined. Thus you couldn't even define n-point functions, a crucial element of quantum field theory.

The $V$ part is not how I would choose to phrase it. I would prefer:
$$\Psi : \mathcal{S}\left(M\right) \rightarrow End\left(\mathcal{D}\right)$$
That is a quantum field takes in a Schwartz function and maps it to an (endomorphic) operator on a dense subset of the state space. This mapping is the smearing operation:
$$
\Psi\left(f\right) = \int_{M}{\Psi(x) f(x) dx}
$$
Thus only after being integrated against does $\Psi$ give an operator.

The authors here are sort of representing the smearing in a "point-like" way. They consider $\Psi$ "at a point" to take in an element of the Schwartz function "at a point" to produce an operator. The value of the Schwartz function at a point is an element of some fiber $V^{*}$ on the manifold which is dual to the values of the field $V$ at that point. I don't think this is a good way to go about it because the fields are distributions and thus don't have vector space values at a point.

 

[Bobjoesmith]

Thanks @DarMM, this comment really helps to clear up some of my confusion. I really appreciate the help

 

[DarMM]

@Bobjoesmith I just went over those notes and they're sort of an odd presentation of the Wightman-Gårding axioms. I much prefer the presentation in Streater and Wightman's classic "PCT, Spin and Statistics and all that" Chapter 3.

The Wightman axioms there conceptually are:

1. There is a Hilbert space carrying a (projective) representation of the Poincaré group.
   The generator of translations $P^{\mu}$ has eigenvalues lying in or on the positive light cone.
   There is one invariant state, the vacuum $\Omega$
   
   These are basically the conditions of a Lorentz covariant quantum theory with sensible spectral properties
2. For each Schwartz function $f$ there is a set of self-adjoint operators $\phi_{i}\left(f\right)$ that are endomorphisms of a dense subspace $\mathcal{D}$ of the Hilbert space. $\Omega$ the vacuum is in this subspace. The unitary (projective) representations of the Poincaré group map $\mathcal{D}$ into itself.
   Whenever $\Psi , \Phi \in \mathcal{D}$ then $\left(\Psi, \phi\left(f\right)\Phi\right)$ is a tempered distribution.
   
   This basically states the existence of a set of operator valued distributions. The second part, together with the fact that they and the Poincaré transformations are endomorphisms, establishes the minimal conditions necessary for them to have n-point functions in any given reference frame.
3. The action of the Poincaré group on these operators is equivalent to transforming their components as if they were tensor or spinor fields on Minkowski spacetime:
   $$U\left(a,A\right)\phi_{j}\left(f\right)U\left(a, A\right)^{-1} = \sum S_{jk}\left(A^{-1}\right)\phi_{k}\left(\left\{a, A\right\}f\right)$$
   
   This just establishes that these operators are in fact fields
4. The fields commute or anti-commute at spacelike distances. This just implements locality.

 

[DarMM]

Some follow up points to note.

Separately from these axioms we have the definition of a field theory. The axioms above ensure we are dealing a Lorentz covariant quantum theory which contains field operators. However it does not preclude the existence of operators which cannot be expressed as functions of the field. Thus we have a relativistic quantum theory with fields. (Note there is no agreed upon standard name for these theories. Also called Wightman-Gårding theories)

We then say a quantum field theory is a relativistic quantum theory with fields where the vacuum is cyclic for the fields. Cyclic meaning the space of all states created by arbitrary polynomials of the fields smeared against arbitrary functions acting on the vacuum is dense in the Hilbert space. This then implies all operators are functions of the fields and thus the entire theory is completely specified by the fields.

I'll say a bit about the Wightman axioms (the separate axioms for the expectation values) and how they are equivalent to the Wightman-Gårding axioms above over the coming days. The reconstruction theorem that proves their equivalence allows you to easily see (I think) how interacting theories don't have a Fock structure.

 

[DarMM]

So the Wightman axioms are as follows:

1. We have a set of tempered distributions (that is distributions that possesses Fourier transforms) $W_{n, A, \sigma}$ over $\mathbb{R}^{dn}$, with $d$ the dimension of the spacetime and $n$ the number of points. $A$ is a transformation index associated with coordinate transformations and $\sigma$ other internal indicies. Among these there is one $W_{0,\sigma} = 1$
   
   Obviously this axiom is just stating the existence of the n-point functions. The final bit simply says the vacuum is normalised and a scalar under spatial transformations.
   I'll be very symbolic below just to save time. When a property doesn't depend on the order $n$ or other indices I'll just drop it.
2. $A$ denotes tensor products of Lorentz transformation indicies. Also the distributions obey:
   $$W\left(f\right) = W\left(\Lambda f\right)$$ with $\Lambda$ the appropriate Lorentz transformation on $f$ given $A, n$.
   
   This is just Lorentz covariance of the n-point functions
3. The distributions obey positivity. That is we have:
   $$\sum_{i,j}W_{i + j}\left(f_{i}\otimes f_{j}\right) \geq 0$$
   For any given set of functions $f_{i}$ over $\mathbb{R}^{dn}$
   
   This is what guarantees the Hilbert space structure with a well defined inner product. As we'll see in the reconstruction theorem the n-point functions essentially are equivalent to the inner product and a collection of $f_{i}$ functions will be the elements of the Hilbert space, i.e. quantum states. Together they can be used to build the Hilbert space.
   
   Non-Fock structure will simply come from the $n \geq 3$ n-point functions not being simply sums of products of the two point function.
4. The Fourier transformation of the n-point functions are supported on (order n tensor products of) the forward light cone $\bar{V}^{+,n}$
   
   This is simply positivity of energy and sensible spectral conditions.
5. We have:
   $$W_{n}\left(x_{1}\cdots x_{j}, x_{j+1}, \cdots x_{n}\right) = \pm W_{n}\left(x_{1}\cdots x_{j+1}, x_{j}, \cdots x_{n}\right)$$
   for $x_{j}$ and $x_{j+1}$ spacelike separated.
   
   This is locality. It will be used to prove some interesting theorems like the Spin and Statistics theorem. Ultimately the Spin and Statistics theorem arises from the fact that if you perform the above exchange for spacelike points with the wrong sign (i.e. the one that doesn't match their spin) and then perform a Lorentz transformation you'll get the identity $W_{n} = -W_{n}$ and thus the field theory vanishes.
6. For spacelike $a$ we have:
   $$
   \lim_{a\rightarrow\infty}W_{n}\left(x_{1}\cdots x_{j}, x_{j+1} + a, \cdots x_{n} + a\right) = W_{n}\left(x_{1}\cdots x_{j}\right)W_{n}\left(x_{j+1}, \cdots x_{n}\right)
   $$
   
   This is the cluster decomposition property and implies uniqueness of the vacuum.