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Physics
616 Fields I
(Spring 2026)
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Room: SRN-385
Time:
Tuesday & Friday,
12:10-1:30 pm
Instructor: Sergei
Lukyanov
office: Serin E364
office phone: (848) 445-9060
e-mail:
sergei@physics.rutgers.edu (preferred)
Office hours: Monday
11:00 am -1:00 pm
Extra meetings can be held by appointment.
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Lecture Notes:
Posted throughout the semester, either before or after each lecture.
They are primarily based on the lecture materials from A. B. Zamolodchikov’s Fields I course. The course will not follow any single textbook closely.
The main reference text:
M. Peskin and D. Schroeder, "An Introduction to Quantum Field Theory"
Homework:
(1) There will one homework per 1-2 weeks.
(2) Late homework
will not be accepted.
The absolute
cutoff time for
homework is 7 pm due date.
(3) Ideally, solutions should be typed
(in LaTeX), but
handwritten solutions are acceptable as long as they are
clearly written. I'll not accept sloppy solutions.
(4) Homeworks
will be graded and give
50%
contribution
to your final
grade.
Exercises:
Occasional "Exercises" will be suggested. They are not graded but strongly recommended.
Exams: No midterm. A take-home Final Exam will be posted near the end of the semester;
at least one week will be given to complete it. Submission rules will match those for homework.
Final grade:
Score % = 50% Homework
+ 40% Final+10% Instructor’s discretion
Grades will be announced a few days before the official deadline so corrections can be requested if needed.
Students with Disabilities:
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If you have a disability, it is essential that you speak to the course supervisor early in the semester to make the necessary arrangements to support a successful learning experience. Also, you must arrange for the course supervisor to receive a Letter of Accommodation from the Office of Disability Services.
http://www.physics.rutgers.edu/ugrad/disabilities.html |
Download the course
info in PDF format
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Students must have a solid background in Classical Mechanics (507),
Electromagnetism (503/504), with particular emphasis on Special Relativity from 504),
as well as Quantum Mechanics (501/502) and Statistical Mechanics (611).
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PLAN OF LECTURES
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This is a tentative schedule of the topics we plan to cover in the course.
It is subject to change, often
without notice.
Changes may occur in response to the pace at which we cover the material,
individual class interests,
and adjustments to the topics included.
Please use this schedule as
a guide to read ahead in the lecture notes
and the textbook.
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CANONICAL QUNTIZATION OF FREE SCALAR FIELD
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Preliminaries (self study):
Lagrangian formulation of classical mechanics. Noether's theorem in classical mechanics. From classical mechanics to field theory: One and three dimensional crystal.
Relativistically invariant classical field theory. Examples: Klein-Gordon model, Electromagnetic field.
Suggested literature:
Lecture notes
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Classical field theory: Noether's theorem. Energy-mometum tensor. Klein-Gordon field.
Suggested literature:
Lecture notes
Secs. 2.1,2.2 in
[PS]
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Quantum Klein-Gordon theory (I):
Hamiltonian formalism. Quantization of the KG theory.
Suggested literature:
Lecture notes
Sec.2.3 in
[PS]
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Quantum Klein-Gordon theory (II):
Casimir effect in massless KG theory. Euler-Maclaurin formula.
Suggested literature:
Lecture notes
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Quantum Klein-Gordon theory (II):
Spectrum of the energy-momentum operator. Heisenberg field operator. Commutators. Feynman
propagator. Euclidean propagator. Wick rotation.
Suggested literature:
Lecture notes
Sec.2.4 in
[PS]
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PATH INTEGRAL IN QUANTUM MECHANICS AND FIELD THEORY.
EUCLIDEAN FIELD THEORY
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- Path integral in Quantum Mechanics (I):
Imaginary time path integral. Relation to Classical Statistical Mechanic.
Suggested literature:
Lecture notes
Sec.9.1 in
[PS]
- Path integral in Quantum Mechanics (II):
Correlation functions. Path integral for a relativistic particle.
Suggested literature:
Lecture notes
- Path integral in Field Theory:
Path integral in Klein-Gordon Theory. Wick's Theorem.
φ^4 theory.
Suggested literature:
Lecture notes
Sec. 4.3, 9.2, 9.3 in
[PS]
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PERTURBATION THEORY. EFFECTIVE ACTION
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- Perturbation Theory for φ^4 theory: Feynman rules. Symmetry factors. Vacuum diagrams.
Suggested literature:
Lecture notes
Sec. 4.1-4.4, 9.2, 9.3 in
[PS]
- Diagramology: Generating Functional. Connected correlations functions. Proper vertices (one-particle irreducible correlation functions).
Suggested literature:
Lecture notes
Sec. 4.1-4.4, 9.2, 9.3 in
[PS]
- Effective Action (I): Definition. Legandre transform. Semiclassical (loop) expansion. The leading (tree) approximation. Tree diagrams.
Suggested literature:
Lecture notes
Sec.11.3, 11.5 in
[PS]
- Effective Action (II): The first order (one-loop) correction. Effective Action vs Generating Functional.
Suggested literature:
Lecture notes
Sec.11.3, 11.5 in
[PS]
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RENORMALIZED PERTURBATION THEORY
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- Evaluation of the diagram contributions (examples): Momentum-space Feynman rules. Leading contribution to
the self-energy. Mass renormalization. Divergences in φ^4 theory. Primitive divergences.
Suggested literature:
Lecture notes
- Renormalization program: Primitive divergences in φ^4 theory. Renormalized perturbation theory.
Divergences in scalar theories.
Suggested literature:
Lecture notes
Sec. 10.1, 10.2 in
[PS]
- Systematics of Renormalization (I): Regularization methods (lattice, proper-time regularization, Pauli-Villars,
dimensional regularizations). Renormalization schemes. Normalization conditions.
Suggested literature:
Lecture notes
Sec. 7.5 in
[PS]
- Systematics of Renormalization (II): Renormalization at the leading (one loop) order.
Renormalization of composite fields in φ^4 theory. Two loop analysis.
Suggested literature:
Lecture notes
Sec. 10.2, 10,4, 10.5 in
[PS]
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RENORMALIZATION GROUP
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- Perturbative Renormalization Group (I): Finite renormalizations. Short distance problem. Massless φ^4 theory.
Renormalization as the scale transformation.
Callan-Symanzik equation.
Suggested literature:
Lecture notes
Sec. 12.2-12.5 in
[PS]
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- Perturbative Renormalization Group (II): Renormalization Group flow. Mass perturbation.
Suggested literature:
Lecture notes
Sec. 12.2-12.5 in
[PS]
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- Second order phase transitions: Landau theory.
Fluctuation theory of phase transition. Landau-Ginzburg
model.
Suggested literature:
Lecture notes
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- Wilson's Renormalization Group (I): Wilson's RG transformation. Wilson's RG flow equation. RG trajectory.
Suggested literature:
Lecture notes
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- Wilson's Renormalization Group (II): Topological properties of RG flow. Fixed points. Scale invariance. Anomalous scale dimensions.
Irrelevant,marginal,relevant fields. Marginally irrelevant and relevant fields. Gaussian fixed point.
Suggested literature:
Lecture notes
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- Wilson's Renormalization Group (III): RG flow and Poincaré-Dulac normal form. Callan-Symanzik equation revisited.
Fixed Points and Criticality. Universality of critical singularities. Scaling relations.
Suggested literature:
Lecture notes
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- Wilson's Renormalization Group (IV): Wilson-Fisher fixed point. ϵ-expansion. O(N) φ^4 theory. Goldstone bosons.
Suggested literature:
Lecture notes
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- Nonlinear Sigma Model at d=2+ϵ: Asymptotic freedom.
Suggested literature:
Lecture notes
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ENERGY-MOMENTUM IN QFT
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- RG and Energy-Momentum Tensor: Conformal transformations. Ward identities. Conformal invariance at the fixed-point.
Scale invariance and density of states.
Suggested literature:
Lecture notes
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- Operator Formalism vs Functional Integral in QFT:
Correlation functions and Energy-Momentum. Space of states. Reflection positivity.
Suggested literature:
Lecture notes
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- Energy-Momentum operators: Stationary States. Energy Positivity. Spectrum of Energy-Momentum. Källén–Lehmann spectral representation.
Suggested literature:
Lecture notes
Sec. 7.1 in
[PS]
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S-MATRIX
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- LSZ formalizm: In- and Out states. Definition of the S-matrix. The LSZ reduction formula.
Suggested literature:
Lecture notes
Sec. 7.2 in
[PS]
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- Fundamental properties of the S-matrix: Crossing symmetry. Unitarity. Optical theorem.
Suggested literature:
Lecture notes
Sec. 7.3 in
[PS]
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Homeworks and
Solutions
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The assignments and solutions are stored in PDF format.
The absolute
cutoff time for homework is 7pm due date.
I'll not accept sloppy solutions. |
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Assigned
on
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Assignment
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Due
Date
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Solutions
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| 1. |
Jan.20,
2026 |
pdf
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Jan.30,
2026 |
pdf
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| 2. |
Jan.20,
2026
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pdf
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Feb.10,
2026 |
pdf
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| 3. |
Jan.20,
2026 |
pdf
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Feb.20,
2026 |
pdf
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| 4. |
Jan.30,
2026 |
pdf
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Mar.03, 2026 |
pdf
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| 5. |
Jan.30,
2026 |
pdf
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Mar.13, 2026 |
pdf
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-->
Spring Recess:
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March 14-22, 2026
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| 6. |
Mar., 2026
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pdf
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Mar.27, 2026
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pdf
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| 7. |
Mar., 2026
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pdf
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Apr.7, 2026
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pdf
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| 8. |
Mar., 2026
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pdf
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Apr.17, 2026
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pdf
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| 9. |
Apr., 2026
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pdf
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Apr.28, 2026
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pdf
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Final:
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The exam must be submitted no later than March 11, 2026
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Grades:
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Final grades
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