Quantum Field Theory


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The Why, What and How

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Read this book on SpringerLink. The particles describing reality could no longer be described solely as particle-like. Instead, they had elements of both waves and particles, and behaved according to a novel set of rules. An illustration between the inherent uncertainty between position and momentum at the quantum level. Heisenberg uncertainty shows up in places where people often least expect it.

Initially, these descriptions troubled physicists a great deal.

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These troubles didn't simply arise because of the philosophical difficulties associated with accepting a non-deterministic Universe or an altered definition of reality, although certainly many were bothered by those aspects. Instead, the difficulties were more robust. The theory of special relativity was well-understood, and yet quantum mechanics, as originally developed, only worked for non-relativistic systems.

Trajectories of a particle in a box also called an infinite square well in classical mechanics A In A , the particle moves at constant velocity, bouncing back and forth. In B-F , wavefunction solutions to the Time-Dependent Schrodinger Equation are shown for the same geometry and potential. The horizontal axis is position, the vertical axis is the real part blue or imaginary part red of the wavefunction. B,C,D are stationary states energy eigenstates , which come from solutions to the Time-Independent Schrodinger Equation.

E,F are non-stationary states, solutions to the Time-Dependent Schrodinger equation. Note that these solutions are not invariant under relativistic transformations; they are only valid in one particular frame of reference. This was the first existential crisis to face quantum physics.

What is Quantum Field Theory?

We say that a theory is relativistically invariant if its laws don't change for different observers: for two people moving at different speeds or in different directions. Formulating a relativistically invariant version of quantum mechanics was a challenge that took the greatest minds in physics many years to overcome, and was finally achieved by Paul Dirac in the late s. Different frames of reference, including different positions and motions, would see different laws The fact that we have a symmetry under 'boosts,' or velocity transformations, tells us we have a conserved quantity: linear momentum.

This is much more difficult to comprehend when momentum isn't simply a quantity associated with a particle, but is rather a quantum mechanical operator.

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The result of his efforts yielded what's now known as the Dirac equation, which describes realistic particles like the electron, and also accounts for:. This was a great leap forward, and the Dirac equation did an excellent job of describing many of the earliest known fundamental particles, including the electron, positron, muon, and even to some extent the proton, neutron, and neutrino. A Universe where electrons and protons are free and collide with photons transitions to a neutral The scattering between electrons and electrons, as well as electrons and photons, can be well-described by the Dirac equation, but photon-photon interactions, which occur in reality, are not.

But it couldn't account for everything. Photons, for instance, couldn't be fully described by the Dirac equation, as they had the wrong particle properties.

Annotated Physics Encyclopædia: Quantum Field Theory

Electron-electron interactions were well-described, but photon-photon interactions were not. Explaining phenomena like radioactive decay were entirely impossible within even Dirac's framework of relativistic quantum mechanics.

Even with this enormous advance, a major component of the story was missing. The big problem was that quantum mechanics, even relativistic quantum mechanics, wasn't quantum enough to describe everything in our Universe. If you have a point charge and a metal conductor nearby, it's an exercise in classical physics alone In quantum mechanics, we discuss how particles respond to that electric field, but the field itself is not quantized as well.

This seems to be the biggest flaw in the formulation of quantum mechanics. Think about what happens if you put two electrons close to one another.

In fact, we often continue to teach this in universities where we explain that quarks and electrons form the lego-bricks from which all matter is made. But this statement hides a deeper truth. According to our best laws of physics, the fundamental building blocks of Nature are not discrete particles at all. Instead they are continuous fluid-like substances, spread throughout all of space.

We call these objects fields. The most familiar examples of fields are the electric and magnetic field. The ripples in these fields give rise to what we call light or, more generally, electromagnetic waves. The field emerging from a magnet is shown on the right.

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If you look closely enough at electromagnetic waves, you'll find that they are made out of particles called photons. The ripples of the electric and magnetic fields get turned into particles when we include the effects of quantum mechanics.


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But this same process is at play for all other particles that we know of.

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