The quantum theory of the fields is the application of the concepts of the Quantum physics to the fields. Exit of the Mechanical quantum relativist, whose interpretation as theory describing only one particle had proven to be incoherent, this theory provides a conceptual framework largely used in Physique of the particles, Physique of the condensed matter and in Physique statistics.

The first quantum theory of the fields to have been born, the quantum electrodynamic , is in XXIe century one of the theories physics having more nice success in its confrontation with the experimental results within the framework of the standard Modèle, in particular thanks to the agreement of the theory with the measurement of utmost precision of the Constante of fine structure.

History

See also: History of the quantum theory of the fields

The theory is born in 1927 with the article founder from the quantum electrodynamic by Dirac: the Quantum theory of the emission and the absorption of the radiation. The formalism then is developed and discussed in the years 1930 by the theorists, those encountering a recurring problem: systematic appearance the infinite ones during calculations of measurable and finished supposed physical sizes being. This difficulty was entirely overcome only in 1948 with the invention of a systematic procedure, the Renormalization, had mainly with Tomonaga, Schwinger and Feynman.

Successes of the quantum electrodynamic , theory of abelian gauge, led the theorists of the years 1960 and 1970 to apply the concepts of this theory to the nonabelian theories of gauge, finally giving rise to current the standard model of the physics of the particles.

In addition, Kadanoff introduced at the end of the years 1960 the idea that the transitions from phases described by statistical physics presented properties of universality and Invariance of scales. Wilson had then the idea to apply the methods of renormalization of the quantum theory of the fields to the description of the critical phenomena.

Quantum fields

Concept of field quantum

The way in which the theory of the fields was introduced by Dirac starting from the elementary particles is known for historical reasons under the name of Second quantification.

• the fields are not related to the Dualité wave-corpuscle.

The elementary particles have already this duality in the acceptance of the term of traditional mechanics. What one understands by field is a concept which allows the creation or the annihilation of particles in any point of space. Like any quantum system, a quantum field has a Hamiltonien and obeys the equation of Schrödinger:

$H\; \backslash \; left|\; \backslash \; psi\; (T)\; \backslash \; right\; \backslash \; rangle\; =\; I\; \backslash \; hbar\; \{\backslash \; partial\; \backslash \; over\; \backslash \; partial\; T\}\; \backslash \; left|\; \backslash \; psi\; (T)\; \backslash \; right\; \backslash \; rangle$

(In theory of the fields, the Lagrangian formalism is easier to use than its Hamiltonian equivalent.)

• With the Second quantification, the indiscernibility of the particles is expressed in terms of number of occupation.

Let us suppose that NR = 3 , with a particle in the state φ₁ and two in the state φ₂, then the function of wave is:

$$

\ frac {1} {\ sqrt {3}} \ left \ phi_1 (r_1) \ phi_2 (r_2) \ phi_2 (r_3) + \ phi_2 (r_1) \ phi_1 (r_2) \ phi_2 (r_3) + \ phi_2 (r_1) \ phi_2 (r_2) \ phi_1 (r_3) \ right]

whereas with the second quantification, this function is simply

$|1,\; 2,0,0,\; \backslash \; cdots\; \backslash \; rangle$

Though the difference is tiny, the second makes it possible to express easily operators creation and annihilation , which adds or removes particles with the state. These operators are very similar to those defined by a Oscillateur quantum harmonic which, in quantum mechanics, creates or destroys quanta of energy.

For example, the operator a₂ with the following effect:

$a\_2\; |\; 1,2,0,0,\; \backslash \; cdots\; \backslash \; rangle\; \backslash \; equiv\; |\; 1,1,0,0,\; \backslash \; cdots\; \backslash \; rangle\; \backslash \; sqrt\; \{2\}$

$a\_2\; |\; 1,1,0,0,\; \backslash \; cdots\; \backslash \; rangle\; \backslash \; equiv\; |\; 1,0,0,0,\; \backslash \; cdots\; \backslash \; rangle$

$a\_2\; |\; 1,0,0,0,\; \backslash \; cdots\; \backslash \; rangle\; \backslash \; equiv\; \backslash \; quad\; 0$

(The factor $\backslash \; sqrt\; \{2\}$ standardizes the function of wave.)

Lastly, it is necessary to introduce “the operators of field” of creation or annihilation of a particle in a point of space.

Just as for only one particle the function of wave is expressed with its kinetic moment, in the same way the operators of field can express themselves using the transformed of Fourier.

For example, $\backslash \; phi\; (\backslash \; mathbf\; \{R\})\; \backslash \; equiv\; \backslash \; sum\_\; \{I\}\; e^\; \{I\; \backslash \; mathbf\; \{K\}\; \_i\; \backslash \; cdot\; \backslash \; mathbf\; \{R\}\}\; a\_\; \{I\}$, which one should not confuse with a function of wave, is the operator of field of boson annihilation.

The Hamiltonians, in physics of the particles, are written like a sum of operators creation and annihilation of field:

$H\; =\; \backslash \; sum\_k\; E\_k\; \backslash ,\; a^\; \backslash \; dagger\_k\; \backslash ,\; a\_k$

That expresses a field of Boson S free, where E_k is the kinetic energy. This Hamiltonian is used to describe Phonon S.

Localization

The experimenter who records a “click” in his detector would like to connect this event, which he interprets like the detection of a “particle” relatively quite localized in space (and time), with the quantum field and his excitations, which leads to the problem of the localization in relativistic quantum physics. For certain types of “particles”, the Opérateur of position of Newton-Wigner brings brief replies.

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