[Physics Class Notes] on Quantum Field Theory Pdf for Exam

In theoretical Physics, Quantum field theory (QFT), incorporates the classical field theory, the theory of quantum mechanics, and the theory of special relativity. The combination of these theories explains the behaviour of subatomic particles and their interactions through various force fields.

The two examples of modern QFT are Quantum Electrodynamics and Quantum Chromodynamics.

In particle physics, Quantum field theory uses the physical models of sub-atomic particles and condensed matter physics to establish models of quasiparticles.

On this page, we will understand what is quantum field, all about the quantum field theory, and the difference between quantum mechanics and quantum field theory.

What is the Quantum Field?

For understanding the quantum field theory, we’ll start with the quantum field.

Quantum field is a quantum-theoretical generalization of classical fields. the 2 archetypal classical fields are:

Maxwell’s electromagnetic field and
Einstein’s metric field of gravitation

A method to believe the method of quantization is that we first reformulate the (still classical) field equations in terms of mathematical operators replacing some numerical quantities (this part is pure algebra/calculus, with no introduction of physics).

On the other hand, we “solve” the resulting operator-valued equations, including solutions that don’t appear within the classical theory.

We also make the assertion (validated by observation) that these new, “nonsensical” (in imagination, not during a mathematical sense) solutions accurately describe nature, including all the observed quantum behaviour that contradicts the classical theory.

What is Quantum Field Theory?

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There are several rationales for employing a quantum field theory.

First, QFT is a basic generalization of classical field theories, which are our most successful non-quantum/ natural theories.
Second, a scientific theory can account for the observed, well-studied creation and destruction of particles, processes that do not exist in physics.
Third, a scientific theory is innately relativistic, and “magically” (not really, just elegant math) resolves problems with causality that plague even relativistic quantum particle theories.

But no, quantum fields don’t interact with matter. Quantum fields are matter during a quantum theory, what we perceive as particles are excitations of the quantum field itself.

Theory of Quantum Mechanics

The term “quantum mechanics” is usually utilized in a minimum of two distinct ways.

It often refers to the overall structure of all “quantum” theories – during which observables correspond to self-adjoint operators on a Hilbert space, and changes of the observer’s point of view (like time evolution) refer to unitary operators.

In terms of this usage, quantum mechanics isn’t such a lot different as just a special case of quantum field theory. Now, let us understand the difference between the two in a tabular format:

Difference Between Quantum Mechanics and Quantum Field Theory

S.No.	Quantum Mechanics	Quantum Field Theory
1.	In QM, the operators are the variables of position and momentum.	In QFT, the operators are numbers.
2.	In QM, time and position enter the wave equation in the following two ways: First derivative w.r.t time Second derivative w.r.t position	No derivatives, both time and position are on the same footing.
3.	In QM the physical variables are quantized (first quantization).	In QFT, the conjugate momentum fields are quantized (second quantization).
4.	Quantum mechanics deals with particles. A particle can have a state which state can vary consistent with the Schrödinger equation for nonrelativistic particles. There are other relativistic equations also, like the Dirac equation. all of them treat particles as fundamental and indestructible.	Quantum theory may be a quantum mechanical theory that’s a theoretical framework combining classical theory, a special theory of relativity, and quantum physics. It is wont to build physical models of subatomic particles (in reference to particle physics) and quasi-particles (in reference to condensed matter physics). during this theory, fields with quantized normal modes of oscillation represent particles.
5.	Quantum mechanics is ok when particles are indestructible.	QFT can affect the creation and destruction of particles that occur in high-energy collisions.

Field Theory

A field theory essentially explains all the physical phenomena in terms of a field and the way in which it interacts with the matter/fields.

For instance, Euclidean field theory is considered a very useful tool for the study of quantum field theory.

Where the Euclidean Quantum Field Theory talks of the relativistic quantum field theory in which time is supplanted by a purely formal imaginary time, causing the replacement of Lorentz covariance by the Euclidean group covariance.

So, we encountered the two terms, i.e., Lorentz covariance and the Euclidean group covariance. Now, we will understand these two terms.

1. Lorentz Group Covariance:

In relativistic physics, Lorentz symmetry, named after Hendrik Lorentz, is an equivalence of observation or observational symmetry thanks to the special theory of relativity implying that the laws of physics stay an equivalent for all observers that are moving with reference to each other within an inertial reference frame

2. Euclidean Group Covariance:

Position operators (p.o.) for relativistic fundamental quantum systems are constructed as operator-valued integrals with reference to Euclidean systems of covariance (ESC), i.e., positive operator-valued (POV) measures being covariant under the Euclidean group, and are expressed in terms of the generators of the inhomogenous Lorentz Transformation/Lorentz Tensor.

This p.o. is partly well-known within the literature where
it is found by other methods.

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