Physics:Quantum 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

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. John Wiley & Sons. 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 (11) ↑ Back to index

1. Physics:Quantum basics

2. Physics:Quantum Postulates

3. Physics:Quantum Hilbert space

4. Physics:Quantum Observables and operators

5. Physics:Quantum mechanics

6. Physics:Quantum mechanics measurements

7. Physics:Quantum state

8. Physics:Quantum system

9. Physics:Quantum superposition

10. Physics:Quantum probability

11. Physics:Quantum Mathematical Foundations of Quantum Theory

2. Conceptual and interpretations (14) ↑ Back to index

12. Physics:Quantum Interpretations of quantum mechanics

13. Physics:Quantum Wave–particle duality

14. Physics:Quantum Complementarity principle

15. Physics:Quantum Uncertainty principle

16. Physics:Quantum Measurement problem

17. Physics:Quantum Bell's theorem

18. Physics:Quantum Hidden variable theory

19. Physics:Quantum nonlocality

20. Physics:Quantum contextuality

21. Physics:Quantum Darwinism

22. Physics:Quantum A Spooky Action at a Distance

23. Physics:Quantum A Walk Through the Universe

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

25. Physics:Quantum measurement problem

3. Mathematical structure and systems (13) ↑ Back to index

26. Physics:Quantum Density matrix

27. Physics:Quantum Exactly solvable quantum systems

28. Physics:Quantum Formulas Collection

29. Physics:Quantum A Matter Of Size

30. Physics:Quantum Symmetry in quantum mechanics

31. Physics:Quantum Angular momentum operator

32. Physics:Quantum Runge–Lenz vector

33. Physics:Quantum Approximation Methods

34. Physics:Quantum Matter Elements and Particles

35. Physics:Quantum Dirac equation

36. Physics:Quantum Klein–Gordon equation

37. Physics:Quantum pendulum

38. 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.

39. Physics:Quantum Atomic structure and spectroscopy

40. Physics:Quantum Hydrogen atom

41. Physics:Quantum number

42. Physics:Quantum Multi-electron atoms

43. Physics:Quantum Fine structure

44. Physics:Quantum Hyperfine structure

45. Physics:Quantum Isotopic shift

46. Physics:Quantum defect

47. Physics:Quantum Zeeman effect

48. Physics:Quantum Stark effect

49. Physics:Quantum Spectral lines and series

50. Physics:Quantum Selection rules

51. Physics:Quantum Fermi's golden rule

52. 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.

53. Physics:Quantum Wavefunction

54. Physics:Quantum Superposition principle

55. Physics:Quantum Eigenstates and eigenvalues

56. Physics:Quantum Boundary conditions and quantization

57. Physics:Quantum Standing waves and modes

58. Physics:Quantum Normal modes and field quantization

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

60. Physics:Quantum Density of states

61. Physics:Quantum carpet

6. Quantum dynamics and evolution (17) ↑ Back to index

62. Physics:Quantum Time evolution

63. Physics:Quantum Schrödinger equation

64. Physics:Quantum Time-dependent Schrödinger equation

65. Physics:Quantum Stationary states

66. Physics:Quantum Perturbation theory

67. Physics:Quantum Time-dependent perturbation theory

68. Physics:Quantum Adiabatic theorem

69. Physics:Quantum Scattering theory

70. Physics:Quantum S-matrix

71. Physics:Quantum tunnelling

72. Physics:Quantum speed limit

73. Physics:Quantum revival

74. Physics:Quantum reflection

75. Physics:Quantum oscillations

76. Physics:Quantum jump

77. Physics:Quantum boomerang effect

78. Physics:Quantum chaos

7. Measurement and information (9) ↑ Back to index

79. Physics:Quantum Measurement theory

80. Physics:Quantum Measurement operators

81. Physics:Quantum Projective measurement

82. Physics:Quantum POVM

83. Physics:Quantum Weak measurement

84. Physics:Quantum Measurement collapse

85. Physics:Quantum entanglement

86. Physics:Quantum Zeno effect

87. Physics:Quantum limit

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

88. Physics:Quantum information theory

89. Physics:Quantum Qubit

90. Physics:Quantum Entanglement

91. Physics:Quantum Gates and circuits

92. Physics:Quantum Computing Algorithms in the NISQ Era

93. Physics:Quantum Noisy Qubits

94. Physics:Quantum random access code

95. Physics:Quantum pseudo-telepathy

96. Physics:Quantum network

97. Physics:Quantum money

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

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

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

99. Physics:Quantum optics beam splitter experiments

100. Physics:Quantum Ultra fast lasers

101. Physics:Quantum Experimental quantum physics

102. Physics:Quantum optics

103. Template:Quantum optics operators

10. Open quantum systems (9) ↑ Back to index

104. Physics:Quantum Open systems

105. Physics:Quantum Master equation

106. Physics:Quantum Lindblad equation

107. Physics:Quantum Decoherence

108. Physics:Quantum dissipation

109. Physics:Quantum Markov semigroup

110. Physics:Quantum Markovian dynamics

111. Physics:Quantum Non-Markovian dynamics

112. Physics:Quantum Trajectories

11. Quantum field theory (20) ↑ Back to index

Structural dependency map of quantum field theory.

113. Physics:Quantum field theory (QFT) basics

114. Physics:Quantum field theory (QFT) core

115. Physics:Quantum Fields and Particles

116. Physics:Quantum Second quantization

117. Physics:Quantum Harmonic Oscillator field modes

118. Physics:Quantum Creation and annihilation operators

119. Physics:Quantum vacuum fluctuations

120. Physics:Quantum Propagators in quantum field theory

121. Physics:Quantum Feynman diagrams

122. Physics:Quantum Path integral formulation

123. Physics:Quantum Renormalization in field theory

124. Physics:Quantum Renormalization group

125. Physics:Quantum Field Theory Gauge symmetry

126. Physics:Quantum Non-Abelian gauge theory

127. Physics:Quantum Electrodynamics (QED)

128. Physics:Quantum chromodynamics (QCD)

129. Physics:Quantum Electroweak theory

130. Physics:Quantum Standard Model

131. Physics:Quantum triviality

132. Physics:Quantum confinement problem

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

133. Physics:Quantum Statistical mechanics

134. Physics:Quantum Partition function

135. Physics:Quantum Distribution functions

136. Physics:Quantum Liouville equation

137. Physics:Quantum Kinetic theory

138. Physics:Quantum Boltzmann equation

139. Physics:Quantum BBGKY hierarchy

140. Physics:Quantum Relaxation and thermalization

141. Physics:Quantum Thermodynamics

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

142. Physics:Quantum Band structure

143. Physics:Quantum Fermi surfaces

144. Physics:Quantum Semiconductor physics

145. Physics:Quantum Phonons

146. Physics:Quantum Electron-phonon interaction

147. Physics:Quantum Superconductivity

148. Physics:Quantum Topological phases of matter

149. Physics:Quantum well

150. Physics:Quantum spin liquid

151. Physics:Quantum spin Hall effect

152. Physics:Quantum phase transition

153. Physics:Quantum critical point

154. 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.

155. Physics:Quantum Fusion reactions and Lawson criterion

156. Physics:Quantum Plasma (fusion context)

157. Physics:Quantum Magnetic confinement fusion

158. Physics:Quantum Inertial confinement fusion

159. Physics:Quantum Plasma instabilities and turbulence

160. Physics:Quantum Tokamak core plasma

161. Physics:Quantum Tokamak edge physics and recycling asymmetries

162. Physics:Quantum Stellarator

15. Timeline (8) ↑ Back to index

163. Physics:Quantum mechanics/Timeline

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

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

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

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

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

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

170. Physics:Quantum mechanics/Timeline/Quiz

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

171. Physics:Quantum topology

172. Physics:Quantum battery

173. Physics:Quantum Supersymmetry

174. Physics:Quantum Black hole thermodynamics

175. Physics:Quantum Holographic principle

176. Physics:Quantum gravity

177. Physics:Quantum De Sitter invariant special relativity

178. Physics:Quantum Doubly special relativity

179. Physics:Quantum arithmetic geometry

180. Physics:Quantum unsolved problems

181. Physics:Quantum Yang-Mills mass gap

182. Physics:Quantum gravity problem

183. Physics:Quantum black hole information paradox

184. Physics:Quantum dark matter problem

185. Physics:Quantum neutrino mass problem

186. Physics:Quantum matter-antimatter asymmetry problem

References

Author: Harold Foppele

Source attribution: Quantum field theory (QFT) basics