Quantum Field Theory: What Is Field Theory?

Fundamental Premise

Every particle and wave in the universe is just an excitation of a quantum field over all of space and time.


I. Scope
II. Fundamental Results
III. What is a field?
IV. Layman's Terms
V. Uncategorized Remarks
VI. References


* The fundamental physical laws of elementary particles has to involve quantum field theory.
* Complicated interacting systems (many-body problems, condensed matter) at small enough levels require quantum field theory. Specific situations will produce non-fundamental "particles", like artificially two-dimensional systems will make excitations called anyons. Collective excitations such as phonons will act as though they were real particles.
* Quantum gravity might involve making classical general relativity into a quantum field theory.

Fundamental Results

* "Quantum" particles means the excitation of a fundamental field comes in discrete chunks, such as the electron, which have properties such as its charge and spin which are only allowed to have certain specific values (hence "quantized.")
* Quantum electrodynamics is the most accurately tested physical theory, the quantization of the classical electromagnetic field. (Magnetic dipole moment of electron is predicted to 10 significant figures.)
* Each particle is an excitation of the same universal quantum field, so each such particle is identical.
* QFT constrains the symmetry of identical particles, so some have Fermi-Dirac and others Bose-Einsein statistics.
* QFT interactions involve products of operators which create and annihilate particles.
* The possibility of creation and annihilation of virtual particles mediates forces. Virtual particles refer to the transient fluctuations of the field, which temporarily possess some of the properties of particles while they exist.

What is a field?

Fields are a mathematical abstraction that convert a position in spacetime into an object representing the amplitude of something at that point. Classical fields are often represented as fields of force at every point of space, where force as a causal concept is not itself an observable quantity. Quantum mechanics is focused on observables, and energy as a scalar field is generally more useful than force as a vector field. Most quantum mechanics is calculated with classical fields.

* The amplitude of this mathematical object could be a scalar, vector, complex number, spinor, or tensor.
* The field is regarded as a physical entity in its own right, since things like conservation of momentum will require accounting for it. Similar logic is used in string theory to regard "branes" as real objects.
* Classical fields will have measurable wave-like excitations. (e.g. electromagnetic waves, gravitational waves)
* Wave-particle duality in quantum mechanics abolishes the distinction of waves and particles for matter, so quantized particles become the excitation of quantum fields which are the fundamental objects of the universe.
* There may ultimately be only a single unified quantum field, with our fundamental "forces" emerging from symmetry breaking at low energies. "Theories of everything" go further trying to predict the values of fundamental constants.
* Quantum gravity implies the notion of spacetime itself becoming fuzzy or volatile at the smallest distance scales. String theories assume spacetime is flat and Einstein gravity emerges from the large-scale behavior of gravity particles. Loop quantum gravity treats spacetime as fundamentally discretized. Distant gamma ray bursts suggest it is very smooth.

Layman's Terms

Quantum Field Theory

* Quantum field theory: There are energy fields throughout space, the splashes and ripples are particles. (e.g. electromagnetic field -> electrons)
* First quantization: Particles turn out to act like waves. (e.g. electrons -> diffraction)
* Second quantization: Waves turn out to act like particles. (e.g. photon)
* Path integral: Things move in every possible way randomly, but the most ridiculous ways mostly cancelled each other.

Quantum Mechanics

* Quantum: "Now if you don't know, now you know."
* Classical: The part of it that makes sense before you add all of the randomness, unless you try to only use that part, in which case absolutely nothing makes any sense at all. It patronizes you by promising it still makes sense on average.
* Quantum mechanics: You try to describe particles as overlapping waves. They randomly turn out to be particles.
* Wave function: Wave describing how the particles will randomly turn out. You can't see it, and it isn't real.
* Uncertainty Principle: Things are really touchy about being pinned down. Bother them the wrong way, they've jumped to something else. The rest of the time they do not care. Maybe you should just leave it alone, it will be the way you left it.
* Superposition: Waves mix together, they're still waves.
* Wavefunction collapse: Waves mixed with surrounding waves when you splashed it. This is very mysterious.
* Entanglement: Waves mixed together and now you can't tell them apart.
* Bell state: That time you had to use the stupid quadratic formula because it doesn't un-F.O.I.L.? And then the teacher replaced the numbers with y's, so you couldn't even do that anymore? It's still just a circle with weirder symbols.
* EPR Paradox: Your mixed up socks were matched together perfectly before you separated them. You call your friend and say you're psychic. He says it was ghosts. Nobody believes you. The History Channel does a special saying it was ghosts.
* Teleportation: You are not allowed to make copies. You make a copy by destroying the original.
* Degeneracy: "On Wednesdays we wear pink."

Uncategorized Remarks

* Quantum field theories often involve taming divergent integrals, "infinities" that need to be swept under the wrong. Integrals essentially mean adding everything up, and divergent means the summation blows up to infinity. So-called "non-renormalizable" theories such as gravity can be treated validly up to some cut-off which is called an effective field theory.
* Fundamental physical theories are so powerful that they effectively predict the situations where they no longer apply.
* These theories are often treated with variational calculus called the Feynman path integral, which mathematicians do not regard as rigorous. It is worth keeping in mind that mathematicians cannot even prove the existence and uniqueness of the fluid mechanics equations at the level of functional analysis, which is essentially Newton's second law for continuous bodies.
* The "theory" itself will be encoded by the assumption of a mathematical object called the Lagrangian. These will not be unique determinants of the observables, so the class of equivalent such theories are what is called "gauge invariant", an obsolete metaphor regarding length contraction of wire thicknesses originating in the study of general relativity.
* String field theories represent string theories as quantum field theories, but there are aspects to them unique to string theory (e.g. string tension), and there are theories in the quantum field framework unrelated to string theory. One is not a mere subset or special case of the other. Strings are quantum 1-dimensional objects, as opposed to 0-dimensional particles.
* The "minimal" level of mathematics in quantum field theory, broadly speaking, are the highest in any branch of science.
* Quantum field theories are consistent with special relativity, and not-too-extreme conditions represent gravity as a massless spin-2 boson called the graviton. The constraints imposed by relativistic spacetime forces some major results.
* Quantum field theory suggests the vacuum energy density of the universe should be roughly 10 to 120th power higher than it is, which is called the worst prediction ever made. It may or may not be related to the dark energy issue in cosmology, where most of the energy density of the universe is unexplained, though it might be attributable to a slightly positive cosmological constant in general relativity. However, this is not itself an explanation, as the value cannot be predicted.
* The Schrodinger and Dirac equations, the latter of which is a relativistic wave equation for the electron, are derived with the use of classical fields. Quantum field theory uses quantum fields, which makes measurable differences.
* In spite of the philosophy on the subject, the wavefunction need not be regarded as physically real, and quantum field theory implies the wave-like particles are essentially epiphenomena of a more fundamental physical object regardless.
* Spin will naturally arise out of relativistic quantum mechanics, but it would be dubious to call it a relativistic effect since its existence can be assumed (postulated) without problem. Spin is the intrinsic angular momentum of a particle.
* In contrast, the creation and annihilation involving anti-matter makes single-body problems inseparable from multi-body problems, which makes the Schrodinger equation fundamentally incapable of describing its interactions.
* Since quantum field theories will treat very different situations as mathematically equivalent, unusual properties such as Majorana fermions can be studied in condensed matter physics as though fundamental particles existed with those properties. Since particles are the excitation of fields, which are universal, things like magnetic monopoles can be studied.


(1) "Quantum Field Theory for the Gifted Amateur"; Tom Lancaster & Stephen Blundell, Oxford University Press. p. 1-2 (2014)

Up One Level: Quantum Field Theory
Last updated: 11/2/2016