# Quantum fields from cluster-decomposition principle

I would like help proving Weinberg's claim (I've quoted him below) that quantum fields are an unavoidable consequence of merging particle-based quantum mechanics with both Lorentz invariance and the cluster-decomposition principle. I'm hoping it can be done directly using a (time-ordered) Dyson series solution of the Schrodinger equation:

$$\Psi(t,x) = Te^{\frac{-i}{\hbar}\int_{-\infty}^{\infty}H(t')dt'}\Psi_0 = T\sum_{n=0}^\infty \frac{(-i / \hbar)^n}{n!}(\int_{-\infty}^{\infty}H(t')dt')^n \Psi_0$$

The method is going to be to expand the operator $H$ into a sum of creation and annihilation operators, which Weinberg says (below) actually involves no physics and is just a mathematical idea that applies to all operators. Assuming you are comfortable with this, which I am not, there should come a moment when inserting a delta function is unavoidable. Could someone help me with this?

To Motivate the Question:

As a hint that this should be possible, I came across Brown's QFT book where he says that, assuming quantum field theory has been developed, the cluster-decomposition principle is just the factoring $$Z(p_1 + p_2) = Z(p_1)Z(p_2)$$ of probability amplitudes, where $p_1(x)$ and $p_2$ are source functions by which he means $$Z(p) = \int [d \psi] \exp(\tfrac{i}{\hbar}\int d^n x ( \mathcal{L} + p_o \psi))$$

(I don't know why he writes $p_0$ and not $p$, and I do not see how doing something like $p_0 = p_1' + p_2'$ allows for this factorization anyway).

Something about the decomposition should naturally motivate inserting delta functions into that $H$ as a way to ensure the $Z(p_1 + p_2) = Z(p_1)Z(p_2)$ right, and should imply quantum fields, but I do not really see how playing with the $H$ in $\exp (\tfrac{i}{\hbar}\int Hdt)\psi_0$ will lead to anything like $Z(p_1 + p_2) = Z(p_1)Z(p_2)$. Nothing in the derivation of Schrodinger's equation or this Dyson series solution actually tells us whether we are working with particles or fields, which is encouraging. Could someone help me with this?

Origin of the question:

In Weinberg's essay, "What is Quantum Field Theory, and What Did We Think It Is?", he talks about the rationale for QFT:

In the course of teaching quantum field theory, I developed a rationale for it, which very briefly is that it is the only way of satisfying the principles of Lorentz invariance plus quantum mechanics plus one other principle. Let me run through this argument very rapidly. The first point is to start with Wigner’s definition of physical multi-particle states as representations of the inhomogeneous Lorentz group.9 You then define annihilation and creation operators $a(\vec{p}, σ, n)$ and $a^†(\vec{p}, σ, n)$ that act on these states. There’s no physics in introducing such operators, for it is easy to see that any operator whatever can be expressed as a functional of them. The existence of a Hamiltonian follows from time-translation invariance, and much of physics is described by the S-matrix... This should all be familiar. The other principle that has to be added is the cluster decomposition principle, which requires that distant experiments give uncorrelated results.10 In order to have cluster decomposition, the Hamiltonian is written not just as any functional of creation and annihilation operators, but as a power series in these operators with coefficients that (aside from a single momentum-conservation delta function) are sufficiently smooth functions of the momenta carried by the operators.

An interesting consequence:

Is the Lagrangian formulation just a mathematical construct allowing one to actually enact the decomposition of $H$ into creation and annihilation operators and this weird delta function, and nothing more? In other words, there may be other ways to do the same thing.

References:

• Weinberg, Quantum Theory of Fields, Vol. 1, Ch. 4.
• Brown, Quantum Field Theory, Ch. 6
• There is an answer at physicsoverflow.org/27619 – Arnold Neumaier Mar 4 '15 at 8:27
• Mr. Neumaier, can you rewrite what I put? I would hate to seem like I'm taking credit for your work. – Jimmy360 Mar 7 '15 at 12:33
• For what concern the Lagrangian formulation, I suggest you to read some other parts of Weinberg's book, namely chapter 7 and 9.If I have not misunderstood his "rationale" the idea is that Lagrangian formalism is necessary so that 1) it is manifest the Lorentz invariance of the theory, that is there exist operators satisfying the requirements of chapter 3, and 2) it is possible to obtain the momentum off-shell propagator of the free fields in a Lorentz invariant fashion (If you try to do that starting the canonical formalism you will have to add non-covariant terms in the interaction,chapter 6) – giulio bullsaver Mar 7 '15 at 16:09