Quantum Field Theory (PH-537) M.Sc Physics 4th Semester Department of Physics NIT Jalandhar Dr Arvind Kumar

Chapters: Canonical Quantization Klein Gordan Field Dirac Field Gauge Field Interacting Theory and Elementary Processes

Books Recommended: Lectures on Quantum Field Theory by Ashok Das Advanced Quantum Mechanics by Schwabl Quantum Field Theory by Michio Kaku Quantum Field Theory by Mark Srednicki An Introduction to Quantum Field Theory by Peskin Field Quantization by W. Griener

Canonical Quantization Chapter I Canonical Quantization Lecture 1 Books Recommended: Lectures on Quantum Field Theory by Ashok Das Quantum Field Theory by Michio Kaku

Why Quantum Field Theory Quantum Mechanics + Special theory of relativity + concept of fields Single particle relativistic quantum mechanics cannot account for processes in which number and type of particles changes Negative energy solutions were not explained in relativistic quantum mechanics

To overcome difficulties we make transition from wave equation to the concept of fields In single particle mechanics, one quantize the single particle in external classical potential. We deals with the operators corresponding to physical observables and also the wave functions which characterize the state of system In QFT, we deals with the fields and these fields are quantized. Particles are identified as different modes of fields. Fields are treated as operators.

QFT find applications in particle physics, condensed matter physics, statistical mechanics, mathematics etc. It is most successful when interactions are small and can be treated perturbatively e.g. QED in terms of fine structure constant α QED predict anomalous magnetic moment correct to six decimal places.

Dirac theory of electron coupled with electromagnetic field lead to QED Negative energy solutions: concept of holes

Higher order corrections to QED lead to problems of infinites or divergence in integrals. Earlier success of QED were lowest order corrections QED contained integrals which diverge as x or k i.e. negligence of space-time structure at small length scale. Classically electron’s self energy was plagued with divergence. Breakdown of causality.

Renormalization of theory: divergent integrals are absorbed into infinite rescaling of coupling constant and masses. QED describe electromagnetic forces only. Need to modify for other fundamental forces And we have Electroweak theory Quantum Chromodynmics

Notations/Conventions: In three dimension Euclidean space Scalar product: Position Angular momentum Some arbitrary vector

Length of vector In four dimensional space (where space and time are considered on equal footing) any point is represented by four coordinates and also a vector in this space will have four components.

We have two kind of vectors in 4-dim space-time Above vectors are related through metric tensor Contavariant vector Covariant vector

Metric tensors are defined as And are related to each other Metric tensors are symmetric

Defining two arbitrary vectors: The scalar product is defined as Above scalar product is invariant under Lorentz transformation and is called Lorentz scalar. (Prove!)

Length of vector in Minkowski space Length of a vector need not always to be positive as was the case for 3-dim Euclidean space. Note that which is the invariant length of any point from origin (Prove!).

The length between two point infinitesimal close to each other is given by is proper time. Prove above statement!

Space-time region is time-like if (using c =1, natural unit) Space-like if Light-like region if

All future processes takes place in future light cone or forward light cone defined by Contragradient and cogradient vectors are defined as (using c = 1) respectively.

We define Lorentz invariant quadratic operator Known as D’ Alembertian operator as (c =1) Energy and momentum are defined in terms of energy-momentum four vector Using above we define the Lorentz invariant scalar Here we used c = 1, otherwise, we have to use E/c

We know the Einstein relationship Where m is the mass of particle. Last two equations define the mass as Lorentz invariant scalar quantity

We know the operator forms Co-ordinate representation of energy momentum four-vector will be

Four dimension Levi-civita tensor which is ant -symmetric four dimensional tensor and

Natural System of Units ћ = 1 and c =1 length dimensions Energy density: Energy/length3 = mass4