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{{Quantum book backlink|Quantum field theory}}
Quantum field theory basics: particles as excitations of fields, spacetime, Feynman diagrams, and gauge symmetries.<br>Quantum field theory describes particles as excitations of underlying fields, combining quantum mechanics with special relativity to explain particle interactions and fundamental forces.<ref>{{cite book |last=Peskin |first=Michael E. |year=1995}}</ref><div style="float:right; border:1px solid #ccc; padding:4px; background:#fff8dc; margin:0 0 1em 1em; width:420px;">
[[File:Quantum_field_theory_basics1.jpg]]
<div style="font-size:90%;">Overview of key concepts in quantum field theory</div>
</div>

=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.<ref>{{cite book |last=Peskin |first=Michael E. |last2=Schroeder |first2=Daniel V. |title=An Introduction to Quantum Field Theory |publisher=Westview Press |year=1995}}</ref>

=== Definition ===

The '''annihilation operator''' <math>\hat{a}_k</math> removes a particle from a mode with momentum <math>k</math>, while the '''creation operator''' <math>\hat{a}_k^\dagger</math> adds a particle:

<math>
\hat{a}_k |n_k\rangle = \sqrt{n_k}\,|n_k - 1\rangle,
</math>

<math>
\hat{a}_k^\dagger |n_k\rangle = \sqrt{n_k+1}\,|n_k + 1\rangle.
</math>

=== Commutation relations ===

For bosons, the operators satisfy

<math>
[\hat{a}_k, \hat{a}_{k'}^\dagger] = \delta_{kk'}.
</math>

For fermions, they satisfy anticommutation relations:

<math>
\{\hat{a}_k, \hat{a}_{k'}^\dagger\} = \delta_{kk'}.
</math>

=== Field expansion ===

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

<math>
\phi(x) = \sum_k \left( \hat{a}_k u_k(x) + \hat{a}_k^\dagger u_k^*(x) \right).
</math>

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.<ref>{{cite book |last=Peskin |first=Michael E. |last2=Schroeder |first2=Daniel V. |title=An Introduction to Quantum Field Theory |publisher=Westview Press |year=1995}}</ref>

=== Definition ===

Fock space is defined as

<math>
\mathcal{F} = \mathbb{C} \oplus \mathcal{H} \oplus (\mathcal{H} \otimes \mathcal{H}) \oplus \cdots,
</math>

where <math>\mathcal{H}</math> 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 <math>|0\rangle</math> contains no particles and satisfies

<math>
\hat{a}_k |0\rangle = 0.
</math>

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

=== Multi-particle states ===

A general multi-particle state is constructed as

<math>
|n_1, n_2, \dots\rangle = \prod_k \frac{(\hat{a}_k^\dagger)^{n_k}}{\sqrt{n_k!}} |0\rangle.
</math>

This describes a configuration with <math>n_k</math> 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: <math>n_k = 0 \text{ or } 1</math>)

=== 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.<ref>{{cite book |last=Peskin |first=Michael E. |last2=Schroeder |first2=Daniel V. |title=An Introduction to Quantum Field Theory |publisher=Westview Press |year=1995}}</ref>

=== Definition ===

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

<math>
D(x - y) = \langle 0 | T\{\phi(x)\phi(y)\} | 0 \rangle,
</math>

where <math>T</math> denotes time ordering.

This quantity represents the amplitude for a particle created at point <math>y</math> to be annihilated at point <math>x</math>.

=== Momentum-space form ===

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

<math>
\tilde{D}(p) = \frac{1}{p^2 - m^2 + i\epsilon}.
</math>

The small imaginary term <math>i\epsilon</math> 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 diagram|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.<ref>{{cite book |last=Peskin |first=Michael E. |last2=Schroeder |first2=Daniel V. |title=An Introduction to Quantum Field Theory |publisher=Westview Press |year=1995}}</ref>

=== 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 ===
[[File:Electron-positron-scattering.svg|thumb|300px|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:

<math>
p^2 \ne m^2.
</math>

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===
* {{cite book|
last1=Pais|first1=A.|author-link1=Abraham Pais
|title=Inward Bound: Of Matter and Forces in the Physical World
|edition=reprint
|year=1994
|orig-date=1986
|publisher=[[Oxford University Press]]
|location=Oxford, New York, Toronto
|isbn=978-0-19-851997-3
}}
* {{cite book
|last=Schweber
|first=S. S.
|author-link=S. S. Schweber
|title=QED and the Men Who Made It: Dyson, Feynman, Schwinger, and Tomonaga
|year=1994
|isbn=978-0-691-03327-3
|url=https://archive.org/details/qedmenwhomadeitd0000schw
|publisher=[[Princeton University Press]]
|url-access=registration
}}
* {{cite book
|last=Feynman
|first=R.P.
|year=2001
|orig-date=1964
|title=The Character of Physical Law
|publisher=[[MIT Press]]
|isbn=978-0-262-56003-0
}}
* {{cite book |last=Feynman |first=R.P.
|year=2006 |orig-date=1985
|title=QED: The Strange Theory of Light and Matter
|publisher=Princeton University Press
|isbn=978-0-691-12575-6
}}
* {{cite book
|last=Gribbin
|first=J.
|author=John Gribbin]]
|year=1998
|title=Q is for Quantum: Particle Physics from A to Z
|publisher=[[Weidenfeld & Nicolson]]
|isbn=978-0-297-81752-9
}}
* {{Cite book |title=The Biggest Ideas in the Universe: quanta and fields |last=Carroll |first=Sean |publisher=[[E. P. Dutton]] |year=2024 |isbn=978-0-593-18660-2 |author=Sean M. Carroll}}

=== Introductory texts===

* {{cite book |author=Pierre van Baal |title=A Course in Field Theory |url=https://www.taylorfrancis.com/books/oa-mono/10.1201/b15364/course-field-theory-pierre-van-baal |isbn=978-0-429-07360-1 |year=2016 |publisher=[[CRC Press]] |doi=10.1201/b15364 }}
* {{cite book
|last1=Cabibbo
|first1=Nicola
|last2=Maiani
|first2=Luciano
|last3=Benhar
|first3=Omar
|title=An Introduction to Gauge Theories
|publisher=[[CRC Press]]
|year=2025
|doi=10.1201/9781003560708
|isbn=9781003560708
|url=https://doi.org/10.1201/9781003560708
}}
* {{cite book |last=Frampton |first=P.H. |author-link=Paul Frampton
|year=2000
|title=Gauge Field Theories
|edition=2nd
|series=Frontiers in Physics
|publisher=[[John Wiley & Sons|Wiley]]
}}; {{cite book|title=2008, 3rd edition|url=https://books.google.com/books?id=AwhkM6hVj-wC|isbn=978-3-527-40835-1 | last1=Frampton | first1=Paul H. | date=22 September 2008 | publisher=John Wiley & Sons }}
* {{cite book |last1=Greiner |first1=W. |author-link=Walter Greiner |last2=Müller |first2=B.|author-link2=Berndt Müller
|year=2000
|title=Gauge Theory of Weak Interactions
|publisher=[[Springer Science+Business Media|Springer]]
|isbn=978-3-540-67672-0
}}
* {{cite book |last1=Itzykson |first1=C. |author-link=Claude Itzykson |last2=Zuber |first2=J.-B.|author-link2=Jean-Bernard Zuber
|year=1980
|title=Quantum Field Theory
|url=https://archive.org/details/quantumfieldtheo0000itzy
|url-access=registration
|publisher=[[McGraw-Hill]]
|isbn=978-0-07-032071-0
}}
* {{cite book |last=Kane |first=G.L. |author-link=Gordon L. Kane
|year=1987
|title=Modern Elementary Particle Physics
|publisher=[[Perseus Books Group|Perseus Group]]
|isbn=978-0-201-11749-3
}}
* {{cite book
|last1=Kleinert
|first1=H.
|author-link=Hagen Kleinert
|last2=Schulte-Frohlinde
|first2=Verena
|year=2001
|title=Critical Properties of φ<sup>4</sup>-Theories
|url=http://users.physik.fu-berlin.de/~kleinert/re.html#B6
|publisher=[[World Scientific]]
|isbn=978-981-02-4658-7
|archive-date=2012-07-22
|access-date=2009-07-02
|archive-url=https://web.archive.org/web/20120722013148/http://users.physik.fu-berlin.de/~kleinert/re.html#B6
}}
* {{cite book
|last=Kleinert
|first=H.
|year=2008
|title=Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation
|url=http://users.physik.fu-berlin.de/~kleinert/public_html/kleiner_reb11/psfiles/mvf.pdf
|publisher=World Scientific
|isbn=978-981-279-170-2
|archive-date=2012-07-16
|access-date=2009-07-02
|archive-url=https://web.archive.org/web/20120716115831/http://users.physik.fu-berlin.de/~kleinert/public_html/kleiner_reb11/psfiles/mvf.pdf
}}
* {{Cite book |last1=Lancaster |first1=Tom
|last2=Blundell |first2=Stephen
|url=https://books.google.com/books?id=Y-0kAwAAQBAJ
|title=Quantum field theory for the gifted amateur
|date=2014
|publisher=Oxford University Press
|isbn=978-0-19-969933-9
|location=Oxford |oclc=859651399}}
* {{cite book |last=Loudon |first=R. |author-link=Rodney Loudon
|year=1983
|title=The Quantum Theory of Light
|publisher=Oxford University Press
|isbn=978-0-19-851155-7
}}
* {{cite book |last1=Maiani |first1=Luciano |author-link1=Luciano Maiani
|last2=Benhar |first2=Omar
|year=2024
|title=Relativistic Quantum Mechanics: An Introduction to Relativistic Quantum Fields
|publisher=CRC Press
|isbn=9781003436263
|url=https://doi.org/10.1201/9781003436263
}}
* {{cite book |last1=Mandl |first1=F. |author-link=Franz Mandl (physicist)
|last2=Shaw |first2=G.
|year=1993
|title=Quantum Field Theory
|publisher=[[John Wiley & Sons]]
|isbn=978-0-471-94186-6
}}
* {{cite book |last=Ryder |first=L.H. |author-link=Lewis Ryder
|year=1985
|title=Quantum Field Theory
|url=https://books.google.com/books?id=nnuW_kVJ500C
|publisher=[[Cambridge University Press]]
|isbn=978-0-521-33859-2
}}
* {{cite book
|last=Schwartz
|first=M.D.
|year=2014
|title=Quantum Field Theory and the Standard Model
|url=http://www.schwartzqft.com
|publisher=Cambridge University Press
|isbn=978-1-107-03473-0
|access-date=2020-05-13
|archive-url=https://web.archive.org/web/20180322014256/http://schwartzqft.com/
|archive-date=2018-03-22
}}
* {{cite book |last=Ynduráin |first=F.J. |author-link=Francisco José Ynduráin
|year=1996
|title=Relativistic Quantum Mechanics and Introduction to Field Theory
|edition=1st
|publisher=Springer
|doi=10.1007/978-3-642-61057-8 |isbn=978-3-540-60453-2
|bibcode=1996rqmi.book.....Y }}
* {{cite book|last1=Greiner|first1=W.|author-link1=Walter Greiner
|last2=Reinhardt|first2=J.
|year=1996
|title=Field Quantization
|url=https://archive.org/details/fieldquantizatio0000grei|url-access=registration|publisher=Springer
|isbn=978-3-540-59179-5
}}
* {{cite book |last1=Peskin |first1=M. |author-link=Michael Peskin |last2=Schroeder |first2=D.
|year=1995
|title=An Introduction to Quantum Field Theory
|url=https://books.google.com/books?id=i35LALN0GosC
|publisher=[[Westview Press]]
|isbn=978-0-201-50397-5
}}
* {{cite book|last=Scharf|first=Günter
|title=Finite Quantum Electrodynamics: The Causal Approach
|edition=third|year=2014
|orig-date=1989
|isbn=978-0-486-49273-5
|publisher=Dover Publications
}}
* {{cite book |last1=Srednicki |first1=M.
|title=Quantum Field Theory
|publisher=[[Cambridge University Press]]
|year=2007
|isbn=978-0521-8644-97
|url=http://www.cambridge.org/us/catalogue/catalogue.asp?isbn=0521864496
}}
* {{cite web|last1=Tong |first1=David |author-link=David Tong (physicist)
|title=Lectures on Quantum Field Theory
|year=2015
|url=https://www.damtp.cam.ac.uk/user/tong/qft.html
|access-date=2016-02-09
}}
* {{cite book
|last=Williams
|first=A.G.
|title=Introduction to Quantum Field Theory: Classical Mechanics to Gauge Field Theories
|volume=
|publisher=[[Cambridge University Press]]
|year=2022
|isbn=978-1-108-47090-2
|url-access=
|url=
}}
* {{cite book
|last1=Zee
|first1=Anthony
|author-link=Anthony Zee
|title=Quantum Field Theory in a Nutshell
|edition=2nd
|year=2010
|isbn=978-0-691-14034-6
|publisher=[[Princeton University Press]]
|url-access=registration
|url=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.
* {{cite book |last=Brown
|first=Lowell S.
|author-link=Lowell S. Brown
|title=Quantum Field Theory
|publisher=[[Cambridge University Press]]
|year=1994
|isbn=978-0-521-46946-3
}}
* {{cite book |last1=Bogoliubov |first1=N.
|last2=Logunov |first2=A.A. |author-link2=Anatoly Logunov
|last3=Oksak |first3=A.I.
|last4=Todorov |first4=I.T.
|year=1990
|title=General Principles of Quantum Field Theory
|publisher=[[Kluwer Academic Publishers]]
|isbn=978-0-7923-0540-8
}}
* {{cite book
|last=Weinberg
|first=S.
|author-link=Steven Weinberg
|title=The Quantum Theory of Fields
|volume=1
|publisher=[[Cambridge University Press]]
|year=1995
|isbn=978-0-521-55001-7
|url-access=registration
|url=https://archive.org/details/quantumtheoryoff00stev
}}
* {{cite book |last1=Badger |first1=Simon |last2=Henn |first2=Johannes |last3=Plefka |first3=Jan Christoph |last4=Zoia |first4=Simone |title=Scattering Amplitudes in Quantum Field Theory |year=2024 |isbn=978-3-031-46987-9 |url=https://link.springer.com/book/10.1007/978-3-031-46987-9 |publisher=Springer |doi=10.1007/978-3-031-46987-9 |bibcode=2024saqf.book.....B }}

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

=References=
{{reflist|3}}

{{Author|Harold Foppele}}

{{Sourceattribution|Quantum field theory (QFT) basics|1}}

Physics:Quantum field theory (QFT) basics - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

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