Physics:Quantum field theory (QFT) basics



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field theory (QFT) basics quantum field theory basics: particles as excitations of fields, spacetime, Feynman diagrams, and gauge symmetries.Quantum field theory describes particles as excitations of underlying fields, combining quantum mechanics with special relativity to explain particle interactions and fundamental forces. Overview of key concepts in quantum field theory Quantum field theory basics: particles as excitations of fields, spacetime, Feynman diagrams, and gauge symmetries.Quantum field theory describes particles as excitations of underlying fields, combining quantum mechanics with special relativity to explain particle interactions and fundamental forces. In quantum field theory, particles are described as excitations of underlying fields. The creation and annihilation operators provide a mathematical framework for adding or removing particles from a system.

Quantum field theory (QFT) basics.

Creation/annihilation operators

In quantum field theory, particles are described as excitations of underlying fields. The creation and annihilation operators provide a mathematical framework for adding or removing particles from a system.

Definition

The annihilation operator ${\hat{a}}_k$ removes a particle from a mode with momentum $k$, while the creation operator ${\hat{a}}_k^{†}$ adds a particle:

${\hat{a}}_k|n_k⟩=\sqrt{n_k}|n_k-1⟩,$

${\hat{a}}_k^{†}|n_k⟩=\sqrt{n_k+1}|n_k+1⟩.$

Commutation relations

For bosons, the operators satisfy

$[{\hat{a}}_k,{\hat{a}}_{k^{\prime }}^{†}]={\delta }_{k{k}^{\prime }}.$

For fermions, they satisfy anticommutation relations:

$\{{\hat{a}}_k,{\hat{a}}_{k^{\prime }}^{†}\}={\delta }_{k{k}^{\prime }}.$

Field expansion

A quantum field can be written as a sum over modes:

$\varphi (x)=\sum _k({\hat{a}}_k{u}_k(x)+{\hat{a}}_k^{†}{u}_k^{*}(x)).$

This expresses the field as a superposition of particle creation and annihilation processes.

Physical significance

Creation and annihilation operators:

• describe particle number changes,
• allow treatment of multi-particle systems,
• form the basis of quantum field quantization.

Fock space

Fock space is the Hilbert space used in quantum field theory to describe systems with a variable number of particles. It is constructed as a direct sum of multi-particle spaces built from single-particle states.

Definition

Fock space is defined as

${ℱ}=\mathbb{ℂ}\oplus {\mathscr{H}}\oplus ({\mathscr{H}}\otimes {\mathscr{H}})\oplus \cdots ,$

where ${\mathscr{H}}$ is the single-particle Hilbert space.

Each term represents states with:

• 0 particles (vacuum),
• 1 particle,
• 2 particles,
• and so on.

Vacuum state

The vacuum state $|0⟩$ contains no particles and satisfies

${\hat{a}}_k|0⟩=0.$

All other states are built by applying creation operators to the vacuum.

Multi-particle states

A general multi-particle state is constructed as

$|n_1,n_2,\dots ⟩=\prod _k\frac{({\hat{a}}_k^{†}{)}^{n_k}}{\sqrt{n_k!}}|0⟩.$

This describes a configuration with $n_k$ particles in each mode.

Bosons and fermions

The structure of Fock space depends on particle statistics:

• Bosons — symmetric states (no restriction on occupation number)
• Fermions — antisymmetric states (Pauli exclusion principle: $n_k=0\text{ or }1$)

Physical significance

Fock space:

• allows description of systems with changing particle number,
• provides the natural setting for quantum fields,
• is essential in particle physics and quantum many-body theory.

Propagators

In quantum field theory, a propagator describes the probability amplitude for a particle to travel from one spacetime point to another. It plays a central role in calculations of particle interactions and quantum processes.

Definition

For a scalar field, the propagator is given by the time-ordered product

$D(x-y)=⟨0|T\{\varphi (x)\varphi (y)\}|0⟩,$

where $T$ denotes time ordering.

This quantity represents the amplitude for a particle created at point $y$ to be annihilated at point $x$.

Momentum-space form

In momentum space, the propagator for a free scalar field has the form

$\tilde{D}(p)=\frac{1}{p^2-m^2+iϵ}.$

The small imaginary term $iϵ$ ensures proper boundary conditions and causality.

Interpretation

Propagators encode how disturbances in a field propagate through spacetime. They are not classical trajectories, but quantum amplitudes that contribute to observable processes.

Role in interactions

In interacting theories, propagators appear as internal lines in Feynman diagrams. They connect interaction vertices and represent virtual particles.

Physical significance

Propagators:

• describe the propagation of particles and fields,
• are fundamental building blocks of quantum field theory,
• enable calculation of scattering amplitudes and correlation functions.

Feynman diagrams

Feynman diagrams are graphical representations of interactions between particles in quantum field theory. They provide an intuitive and systematic way to calculate probability amplitudes for physical processes.

Basic elements

A Feynman diagram consists of:

• External lines — incoming and outgoing particles
• Internal lines — propagators (virtual particles)
• Vertices — interaction points

Each element corresponds to a mathematical expression in perturbation theory.

Example

Example of a Feynman diagram: electron–photon scattering. Straight lines represent fermions, wavy lines represent photons, and vertices represent interactions.

A simple interaction, such as electron–photon scattering, can be represented by a diagram where:

• straight lines represent fermions,
• wavy lines represent photons,
• vertices represent electromagnetic interactions.

Perturbation theory

Feynman diagrams arise from expanding the evolution operator in powers of the interaction strength. Each diagram corresponds to a term in this expansion.

The total amplitude is obtained by summing over all relevant diagrams.

Feynman rules

Each quantum field theory has a set of Feynman rules that translate diagrams into mathematical expressions:

• propagators correspond to internal lines,
• interaction terms determine vertex factors,
• external lines correspond to particle states.

These rules allow systematic computation of scattering amplitudes.

Virtual particles

Internal lines represent virtual particles, which do not satisfy the usual energy–momentum relation:

$p^2\ne m^2.$

They are not directly observable but contribute to measurable quantities.

Physical significance

Feynman diagrams:

• provide a visual representation of particle interactions,
• simplify complex calculations in quantum field theory,
• are essential in particle physics and high-energy experiments.

They are one of the most powerful tools for connecting theory with experimental predictions.

Further reading

General readers

• Pais, A. (1994). Inward Bound: Of Matter and Forces in the Physical World (reprint ed.). Oxford, New York, Toronto: Oxford University Press. ISBN 978-0-19-851997-3. 
• Schweber, S. S. (1994). QED and the Men Who Made It: Dyson, Feynman, Schwinger, and Tomonaga. Princeton University Press. ISBN 978-0-691-03327-3. https://archive.org/details/qedmenwhomadeitd0000schw. 
• Feynman, R.P. (2001). The Character of Physical Law. MIT Press. ISBN 978-0-262-56003-0. 
• Feynman, R.P. (2006). QED: The Strange Theory of Light and Matter. Princeton University Press. ISBN 978-0-691-12575-6. 
• Gribbin, J. (1998). Q is for Quantum: Particle Physics from A to Z. Weidenfeld & Nicolson. ISBN 978-0-297-81752-9. 
• Carroll, Sean (2024). The Biggest Ideas in the Universe: quanta and fields. E. P. Dutton. ISBN 978-0-593-18660-2. 

Introductory texts

• Pierre van Baal (2016). A Course in Field Theory. CRC Press. doi:10.1201/b15364. ISBN 978-0-429-07360-1. https://www.taylorfrancis.com/books/oa-mono/10.1201/b15364/course-field-theory-pierre-van-baal. 
• Cabibbo, Nicola; Maiani, Luciano; Benhar, Omar (2025). An Introduction to Gauge Theories. CRC Press. doi:10.1201/9781003560708. ISBN 9781003560708. https://doi.org/10.1201/9781003560708. 
• Frampton, P.H. (2000). Gauge Field Theories. Frontiers in Physics (2nd ed.). Wiley. ; Frampton, Paul H. (22 September 2008). 2008, 3rd edition. John Wiley & Sons. ISBN 978-3-527-40835-1. https://books.google.com/books?id=AwhkM6hVj-wC. 
• Greiner, W.; Müller, B. (2000). Gauge Theory of Weak Interactions. Springer. ISBN 978-3-540-67672-0. 
• Itzykson, C.; Zuber, J.-B. (1980). Quantum Field Theory. McGraw-Hill. ISBN 978-0-07-032071-0. https://archive.org/details/quantumfieldtheo0000itzy. 
• Kane, G.L. (1987). Modern Elementary Particle Physics. Perseus Group. ISBN 978-0-201-11749-3. 
• Kleinert, H.; Schulte-Frohlinde, Verena (2001). Critical Properties of φ⁴-Theories. World Scientific. ISBN 978-981-02-4658-7. http://users.physik.fu-berlin.de/~kleinert/re.html#B6. Retrieved 2009-07-02. 
• Kleinert, H. (2008). Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation. World Scientific. ISBN 978-981-279-170-2. http://users.physik.fu-berlin.de/~kleinert/public_html/kleiner_reb11/psfiles/mvf.pdf. Retrieved 2009-07-02. 
• Lancaster, Tom; Blundell, Stephen (2014). Quantum field theory for the gifted amateur. Oxford: Oxford University Press. ISBN 978-0-19-969933-9. OCLC 859651399. https://books.google.com/books?id=Y-0kAwAAQBAJ. 
• Loudon, R. (1983). The Quantum Theory of Light. Oxford University Press. ISBN 978-0-19-851155-7. 
• Maiani, Luciano; Benhar, Omar (2024). Relativistic Quantum Mechanics: An Introduction to Relativistic Quantum Fields. CRC Press. ISBN 9781003436263. https://doi.org/10.1201/9781003436263. 
• Mandl, F.; Shaw, G. (1993). Quantum Field Theory. Wiley. ISBN 978-0-471-94186-6. 
• Ryder, L.H. (1985). Quantum Field Theory. Cambridge University Press. ISBN 978-0-521-33859-2. https://books.google.com/books?id=nnuW_kVJ500C. 
• Schwartz, M.D. (2014). Quantum Field Theory and the Standard Model. Cambridge University Press. ISBN 978-1-107-03473-0. http://www.schwartzqft.com. Retrieved 2020-05-13. 
• Ynduráin, F.J. (1996). Relativistic Quantum Mechanics and Introduction to Field Theory (1st ed.). Springer. doi:10.1007/978-3-642-61057-8. ISBN 978-3-540-60453-2. Bibcode: 1996rqmi.book.....Y. 
• Greiner, W.; Reinhardt, J. (1996). Field Quantization. Springer. ISBN 978-3-540-59179-5. https://archive.org/details/fieldquantizatio0000grei. 
• Peskin, M.; Schroeder, D. (1995). An Introduction to Quantum Field Theory. Westview Press. ISBN 978-0-201-50397-5. https://books.google.com/books?id=i35LALN0GosC. 
• Scharf, Günter (2014). Finite Quantum Electrodynamics: The Causal Approach (third ed.). Dover Publications. ISBN 978-0-486-49273-5. 
• Srednicki, M. (2007). Quantum Field Theory. Cambridge University Press. ISBN 978-0521-8644-97. http://www.cambridge.org/us/catalogue/catalogue.asp?isbn=0521864496. 
• Tong, David (2015). "Lectures on Quantum Field Theory". https://www.damtp.cam.ac.uk/user/tong/qft.html. 
• Williams, A.G. (2022). Introduction to Quantum Field Theory: Classical Mechanics to Gauge Field Theories. Cambridge University Press. ISBN 978-1-108-47090-2. 
• Zee, Anthony (2010). Quantum Field Theory in a Nutshell (2nd ed.). Princeton University Press. ISBN 978-0-691-14034-6. https://archive.org/details/isbn_9780691140346. 

Advanced texts

• Umezawa, H. (1956) Quantum Field Theory. North Holland Puplishing.
• Barton, G. (1963). Introduction to Advanced Field Theory. Intescience Publishers.
• Brown, Lowell S. (1994). Quantum Field Theory. Cambridge University Press. ISBN 978-0-521-46946-3. 
• Bogoliubov, N.; Logunov, A.A.; Oksak, A.I.; Todorov, I.T. (1990). General Principles of Quantum Field Theory. Kluwer Academic Publishers. ISBN 978-0-7923-0540-8. 
• Weinberg, S. (1995). The Quantum Theory of Fields. 1. Cambridge University Press. ISBN 978-0-521-55001-7. https://archive.org/details/quantumtheoryoff00stev. 
• Badger, Simon; Henn, Johannes; Plefka, Jan Christoph; Zoia, Simone (2024). Scattering Amplitudes in Quantum Field Theory. Springer. doi:10.1007/978-3-031-46987-9. ISBN 978-3-031-46987-9. Bibcode: 2024saqf.book.....B. https://link.springer.com/book/10.1007/978-3-031-46987-9. 

Index

Core theory

1. Foundations

2. Conceptual and interpretations

3. Mathematical structure and systems

4. Atomic and spectroscopy

5. Wavefunctions and modes

6. Quantum dynamics and evolution

7. Measurement and information

8. Quantum information and computing

Applications and extensions

9. Quantum optics and experiments

10. Open quantum systems

11. Quantum field theory

12. Statistical mechanics and kinetic theory

13. Condensed matter and solid-state physics

14. Plasma and fusion physics

15. Timeline

16. Advanced and frontier topics

Book II

• Matter by scale

Book III

• Methods and tools

Book IV

• Data Analysis Techniques

Full contents

1. Foundations (14) Back to index

1. Physics:Quantum basics

2. Physics:Quantum photoelectric effect

3. Physics:Quantum black-body radiation

4. Physics:Quantum Planck constant

5. Physics:Quantum Postulates

6. Physics:Quantum Hilbert space

7. Physics:Quantum Observables and operators

8. Physics:Quantum mechanics

9. Physics:Quantum mechanics measurements

10. Physics:Quantum state

11. Physics:Quantum system

12. Physics:Quantum superposition

13. Physics:Quantum probability

14. Physics:Quantum Mathematical Foundations of Quantum Theory

2. Conceptual and interpretations (14) Back to index

15. Physics:Quantum Interpretations of quantum mechanics

16. Physics:Quantum Wave–particle duality

17. Physics:Quantum Complementarity principle

18. Physics:Quantum Uncertainty principle

19. Physics:Quantum Measurement problem

20. Physics:Quantum Bell's theorem

21. Physics:Quantum Hidden variable theory

22. Physics:Quantum nonlocality

23. Physics:Quantum contextuality

24. Physics:Quantum Darwinism

25. Physics:Quantum A Spooky Action at a Distance

26. Physics:Quantum A Walk Through the Universe

27. Physics:Quantum The Secret of Cohesion and How Waves Hold Matter Together

28. Physics:Quantum measurement problem

3. Mathematical structure and systems (15) Back to index

29. Physics:Quantum Density matrix

30. Physics:Quantum Exactly solvable quantum systems

31. Physics:Quantum many-body problem

32. Physics:Quantum Formulas Collection

33. Physics:Quantum A Matter Of Size

34. Physics:Quantum Symmetry in quantum mechanics

35. Physics:Quantum Noether theorem

36. Physics:Quantum Angular momentum operator

37. Physics:Quantum Runge–Lenz vector

38. Physics:Quantum Approximation Methods

39. Physics:Quantum Matter Elements and Particles

40. Physics:Quantum Dirac equation

41. Physics:Quantum Klein–Gordon equation

42. Physics:Quantum pendulum

43. Physics:Quantum configuration space

4. Atomic and spectroscopy (14) Back to index

Quantum atomic structure and spectroscopy: orbitals, energy levels, and emission and absorption spectra.

44. Physics:Quantum Atomic structure and spectroscopy

45. Physics:Quantum Hydrogen atom

46. Physics:Quantum number

47. Physics:Quantum Multi-electron atoms

48. Physics:Quantum Fine structure

49. Physics:Quantum Hyperfine structure

50. Physics:Quantum Isotopic shift

51. Physics:Quantum defect

52. Physics:Quantum Zeeman effect

53. Physics:Quantum Stark effect

54. Physics:Quantum Spectral lines and series

55. Physics:Quantum Selection rules

56. Physics:Quantum Fermi's golden rule

57. Physics:Quantum beats

5. Wavefunctions and modes (9) Back to index

A quantum wavefunction showing probability amplitude in space; the square of its magnitude gives the probability density.

58. Physics:Quantum Wavefunction

59. Physics:Quantum Superposition principle

60. Physics:Quantum Eigenstates and eigenvalues

61. Physics:Quantum Boundary conditions and quantization

62. Physics:Quantum Standing waves and modes

63. Physics:Quantum Normal modes and field quantization

64. Physics:Number of independent spatial modes in a spherical volume

65. Physics:Quantum Density of states

66. Physics:Quantum carpet

6. Quantum dynamics and evolution (20) Back to index

67. Physics:Quantum Time evolution

68. Physics:Quantum Schrödinger equation

69. Physics:Quantum Time-dependent Schrödinger equation

70. Physics:Quantum Stationary states

71. Physics:Quantum Perturbation theory

72. Physics:Quantum Time-dependent perturbation theory

73. Physics:Quantum Adiabatic theorem

74. Physics:Quantum Berry phase

75. Physics:Quantum Aharonov-Bohm effect

76. Physics:Quantum Scattering theory

77. Physics:Quantum Scattering matrix

78. Physics:Quantum S-matrix

79. Physics:Quantum tunnelling

80. Physics:Quantum speed limit

81. Physics:Quantum revival

82. Physics:Quantum reflection

83. Physics:Quantum oscillations

84. Physics:Quantum jump

85. Physics:Quantum boomerang effect

86. Physics:Quantum chaos

7. Measurement and information (9) Back to index

87. Physics:Quantum Measurement theory

88. Physics:Quantum Measurement operators

89. Physics:Quantum Projective measurement

90. Physics:Quantum POVM

91. Physics:Quantum Weak measurement

92. Physics:Quantum Measurement collapse

93. Physics:Quantum entanglement

94. Physics:Quantum Zeno effect

95. Physics:Quantum limit

8. Quantum information and computing (10) Back to index

96. Physics:Quantum information theory

97. Physics:Quantum Qubit

98. Physics:Quantum Entanglement

99. Physics:Quantum Gates and circuits

100. Physics:Quantum Computing Algorithms in the NISQ Era

101. Physics:Quantum Noisy Qubits

102. Physics:Quantum random access code

103. Physics:Quantum pseudo-telepathy

104. Physics:Quantum network

105. Physics:Quantum money

9. Quantum optics and experiments (5) Back to index

Experimental quantum physics: qubits, dilution refrigerators, quantum communication, and laboratory systems.

106. Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy

107. Physics:Quantum optics beam splitter experiments

108. Physics:Quantum Ultra fast lasers

109. Physics:Quantum Experimental quantum physics

110. Physics:Quantum optics

111. Template:Quantum optics operators

10. Open quantum systems (9) Back to index

112. Physics:Quantum Open systems

113. Physics:Quantum Master equation

114. Physics:Quantum Lindblad equation

115. Physics:Quantum Decoherence

116. Physics:Quantum dissipation

117. Physics:Quantum Markov semigroup

118. Physics:Quantum Markovian dynamics

119. Physics:Quantum Non-Markovian dynamics

120. Physics:Quantum Trajectories

11. Quantum field theory (23) Back to index

Structural dependency map of quantum field theory.

121. Physics:Quantum field theory (QFT) basics

122. Physics:Quantum field theory (QFT) core

123. Physics:Quantum Fields and Particles

124. Physics:Quantum Second quantization

125. Physics:Quantum Fock space

126. Physics:Quantum Harmonic Oscillator field modes

127. Physics:Quantum Creation and annihilation operators

128. Physics:Quantum vacuum fluctuations

129. Physics:Quantum Casimir effect

130. Physics:Quantum Propagators in quantum field theory

131. Physics:Quantum Feynman diagrams

132. Physics:Quantum Path integral formulation

133. Physics:Quantum Renormalization in field theory

134. Physics:Quantum Renormalization group

135. Physics:Quantum Field Theory Gauge symmetry

136. Physics:Quantum Spontaneous symmetry breaking

137. Physics:Quantum Non-Abelian gauge theory

138. Physics:Quantum Electrodynamics (QED)

139. Physics:Quantum chromodynamics (QCD)

140. Physics:Quantum Electroweak theory

141. Physics:Quantum Standard Model

142. Physics:Quantum triviality

143. Physics:Quantum confinement problem

12. Statistical mechanics and kinetic theory (9) Back to index

144. Physics:Quantum Statistical mechanics

145. Physics:Quantum Partition function

146. Physics:Quantum Distribution functions

147. Physics:Quantum Liouville equation

148. Physics:Quantum Kinetic theory

149. Physics:Quantum Boltzmann equation

150. Physics:Quantum BBGKY hierarchy

151. Physics:Quantum Relaxation and thermalization

152. Physics:Quantum Thermodynamics

13. Condensed matter and solid-state physics (15) Back to index

153. Physics:Quantum Band structure

154. Physics:Quantum Fermi surfaces

155. Physics:Quantum Landau levels

156. Physics:Quantum Semiconductor physics

157. Physics:Quantum Phonons

158. Physics:Quantum Electron-phonon interaction

159. Physics:Quantum Exchange interaction

160. Physics:Quantum Superconductivity

161. Physics:Quantum Topological phases of matter

162. Physics:Quantum well

163. Physics:Quantum spin liquid

164. Physics:Quantum spin Hall effect

165. Physics:Quantum phase transition

166. Physics:Quantum critical point

167. Physics:Quantum dot

14. Plasma and fusion physics (8) Back to index

Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.

168. Physics:Quantum Fusion reactions and Lawson criterion

169. Physics:Quantum Plasma (fusion context)

170. Physics:Quantum Magnetic confinement fusion

171. Physics:Quantum Inertial confinement fusion

172. Physics:Quantum Plasma instabilities and turbulence

173. Physics:Quantum Tokamak core plasma

174. Physics:Quantum Tokamak edge physics and recycling asymmetries

175. Physics:Quantum Stellarator

15. Timeline (8) Back to index

176. Physics:Quantum mechanics/Timeline

177. Physics:Quantum mechanics/Timeline/Pre-quantum era

178. Physics:Quantum mechanics/Timeline/Old quantum theory

179. Physics:Quantum mechanics/Timeline/Modern quantum mechanics

180. Physics:Quantum mechanics/Timeline/Quantum field theory era

181. Physics:Quantum mechanics/Timeline/Quantum information era

182. Physics:Quantum mechanics/Timeline/Quantum technology era

183. Physics:Quantum mechanics/Timeline/Quiz

16. Advanced and frontier topics (16) Back to index

184. Physics:Quantum topology

185. Physics:Quantum battery

186. Physics:Quantum Supersymmetry

187. Physics:Quantum Black hole thermodynamics

188. Physics:Quantum Holographic principle

189. Physics:Quantum gravity

190. Physics:Quantum De Sitter invariant special relativity

191. Physics:Quantum Doubly special relativity

192. Physics:Quantum arithmetic geometry

193. Physics:Quantum unsolved problems

194. Physics:Quantum Yang-Mills mass gap

195. Physics:Quantum gravity problem

196. Physics:Quantum black hole information paradox

197. Physics:Quantum dark matter problem

198. Physics:Quantum neutrino mass problem

199. Physics:Quantum matter-antimatter asymmetry problem

References

Author: Harold Foppele

Source attribution: Physics:Quantum field theory (QFT) basics