As I understand it, the need for quantum field theory (QFT) arises due to the incompatibility between special relativity (SR) and "ordinary" quantum mechanics (QM). By this, I mean that "ordinary" QM has no mechanism to handle systems of varying number of particles, however, special relativity predicts the possibility of particle creation and thus systems with a variable number of particles. "Ordinary" QM also runs into problems such as negative probabilities and a breakdown of causality.(adsbygoogle = window.adsbygoogle || []).push({});

My question though, is whyfields? Why not some other theoretic construction? It makes sense that quantities that were described by classical fields previously, such as the electromagnetism, will be quantum mechanically be described by some sort of "quantum field", but I'm struggling to fully understand the motivation for constructing quantum fields for which particles are emergent quantities (i.e. excitations of their underlying quantum field)?!

I've read Weinberg's paper "What is Quantum Field Theory, and What Did We Think It Is?" , and in it he states that QFT is an inevitable consequence of trying to construct a quantum theory that obeys the principles of SR (namely Lorentz invariance) and also the Cluster Decomposition Principle. Is this sufficient motivation from the start to consider fields?

Here are a couple of ideas I have (although I'm not sure if they're correct at all?!):

1) Is it simply the case that, in order to satisfy Lorentz invariance, interactions must be expressed in terms of density fields, i.e. $$V(t)=\int\,dx\mathcal{H}(t,\mathbf{x})$$ where ##\mathcal{H}(t,\mathbf{x})## is a Lorentz scalar and commutes for spacelike separations, $$\left[\mathcal{H}(t,\mathbf{x}),\mathcal{H}(t,\mathbf{y})\right] =0$$ for ##\left(\mathbf{x}-\mathbf{y}\right)^{2}\geq 0##.

Furthermore, the cluster decomposition principle requires that one constructs ##\mathcal{H}(t,\mathbf{x})## out of creation and annihilation operators, however in order to construct a Lorentz scalar out of such operators they must be "coupled" in some way. The most natural solution to this is to construct ##\mathcal{H}(t,\mathbf{x})## out offields. With this in mind, we are motivated to consider quantum fields as opposed to a particle approach.

2) Since single-particle mechanics is "out of the window" if one includes SR, we need to consider constructing multi-particle states. The most elegant way to do this is to introduce the formalism ofsecond quantisation -instead of asking the question "which particle is in which state", which is meaningless since the particles are indistinguishable, we instead ask "how many particles are in each single-particle state". By doing this we can represent a quantum state of a many-body system in terms of the number of particles in each of the available single-particle states. Furthermore, one can introduce so-calledcreationandannihilationoperators to add and remove particles from each single-particle state, respectively. In this sense, a many-body quantum state can be constructed by acting on the vacuum state of the theory with the creation operator.

Now, in principle, since particles can be created (or annihilated) from the vacuum at any point in spacetime and also should be able to propagate through spacetime continuously, one needs to construct afieldof creation and annihilation operators - a set for each spacetime point - a so-called "quantum field". In this sense, a quantum field acts on the vacuum to create a particle, and since this is the minimum possible excitation of the quantum field, the particlescreated are aquantaof the underlying quantum field.

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# A Quantum Field Theory - Why quantise fields?

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