Exponential object

In quantum field theory, a fermionic field is a quantum field whose quanta are fermions; that is, they obey Fermi–Dirac statistics. Fermionic fields obey canonical anticommutation relations rather than the canonical commutation relations of bosonic fields.

The most prominent example of a fermionic field is the Dirac field, which describes fermions with spin-1/2: electrons, protons, quarks, etc. The Dirac field can be described as either a 4-component spinor or as a pair of 2-component Weyl spinors. Spin-1/2 Majorana fermions, such as the hypothetical neutralino, can be described as either a dependent 4-component Majorana spinor or a single 2-component Weyl spinor. It is not known whether the neutrino is a Majorana fermion or a Dirac fermion (see also Neutrinoless double-beta decay for experimental efforts to determine this).

Basic properties

Free (non-interacting) fermionic fields obey canonical anticommutation relations, i.e., involve the anticommutators {a,b} = ab + ba rather than the commutators [a,b] = ab − ba of bosonic or standard quantum mechanics. Those relations also hold for interacting fermionic fields in the interaction picture, where the fields evolve in time as if free and the effects of the interaction are encoded in the evolution of the states.

It is these anticommutation relations that imply Fermi–Dirac statistics for the field quanta. They also result in the Pauli exclusion principle: two fermionic particles cannot occupy the same state at the same time.

Dirac fields

The prominent example of a spin-1/2 fermion field is the Dirac field (named after Paul Dirac), and denoted by ψ(x). The equation of motion for a free field is the Dirac equation,

$(i\gamma ^{\mu }\partial _{\mu }-m)\psi (x)=0.\,$

where γ^μ are gamma matrices and m is the mass. The simplest possible solutions to this equation are plane wave solutions, $\psi _{1}(x)=u(p)e^{-ip.x}\,$ and $\psi _{2}(x)=v(p)e^{ip.x}\,$. These plane wave solutions form a basis for the Fourier components of ψ(x), allowing for the general expansion of the Dirac field as follows,

$\psi (x)=\int {\frac {d^{3}p}{(2\pi )^{3}}}{\frac {1}{\sqrt {2E_{p}}}}\sum _{s}\left(a_{\textbf {p}}^{s}u^{s}(p)e^{-ip\cdot x}+b_{\textbf {p}}^{s\dagger }v^{s}(p)e^{ip\cdot x}\right).\,$

The a and b labels are spinor indices and the s indices represent spin labels and so for the electron, a spin 1/2 particle, s = +1/2 or s=−1/2. The energy factor is the result of having a Lorentz invariant integration measure. Since ψ(x) can be thought of as an operator, the coefficients of its Fourier modes must be operators too. Hence, $a_{\textbf {p}}^{s}$ and $b_{\textbf {p}}^{s\dagger }$ are operators. The properties of these operators can be discerned from the properties of the field. ψ(x) and $\psi (y)^{\dagger }$ obey the anticommutation relations

$\{\psi _{a}({\textbf {x}}),\psi _{b}^{\dagger }({\textbf {y}})\}=\delta ^{(3)}({\textbf {x}}-{\textbf {y}})\delta _{ab},$

By putting in the expansions for ψ(x) and ψ(y), the anticommutation relations for the coefficients can be computed.

$\{a_{\textbf {p}}^{r},a_{\textbf {q}}^{s\dagger }\}=\{b_{\textbf {p}}^{r},b_{\textbf {q}}^{s\dagger }\}=(2\pi )^{3}\delta ^{3}({\textbf {p}}-{\textbf {q}})\delta ^{rs},\,$

In a manner analogous to non-relativistic annihilation and creation operators and their commutators, these algebras lead to the physical interpretation that $a_{\textbf {p}}^{s\dagger }$ creates a fermion of momentum p and spin s, and $b_{\textbf {q}}^{r\dagger }$ creates an antifermion of momentum q and spin r. The general field ψ(x) is now seen to be a weighed (by the energy factor) summation over all possible spins and momenta for creating fermions and antifermions. Its conjugate field, ${\bar {\psi }}\ {\stackrel {\mathrm {def} }{=}}\ \psi ^{\dagger }\gamma ^{0}$, is the opposite, a weighted summation over all possible spins and momenta for annihilating fermions and antifermions.

With the field modes understood and the conjugate field defined, it is possible to construct Lorentz invariant quantities for fermionic fields. The simplest is the quantity ${\overline {\psi }}\psi \,$. This makes the reason for the choice of ${\bar {\psi }}=\psi ^{\dagger }\gamma ^{0}$clear. This is because the general Lorentz transform on ψ is not unitary so the quantity $\psi ^{\dagger }\psi$ would not be invariant under such transforms, so the inclusion of $\gamma ^{0}\,$ is to correct for this. The other possible non-zero Lorentz invariant quantity, up to an overall conjugation, constructible from the fermionic fields is ${\overline {\psi }}\gamma ^{\mu }\partial _{\mu }\psi$.

Since linear combinations of these quantities are also Lorentz invariant, this leads naturally to the Lagrangian density for the Dirac field by the requirement that the Euler–Lagrange equation of the system recover the Dirac equation.

${\mathcal {L}}_{D}={\bar {\psi }}(i\gamma ^{\mu }\partial _{\mu }-m)\psi \,$

Such an expression has its indices suppressed. When reintroduced the full expression is

${\mathcal {L}}_{D}={\bar {\psi }}_{a}(i\gamma _{ab}^{\mu }\partial _{\mu }-m\mathbb {I} _{ab})\psi _{b}\,$

Given the expression for ψ(x) we can construct the Feynman propagator for the fermion field:

$D_{F}(x-y)=\langle 0|T(\psi (x){\bar {\psi }}(y))|0\rangle$

we define the time-ordered product for fermions with a minus sign due to their anticommuting nature

$T(\psi (x){\bar {\psi }}(y))\ {\stackrel {\mathrm {def} }{=}}\ \theta (x^{0}-y^{0})\psi (x){\bar {\psi }}(y)-\theta (y^{0}-x^{0}){\bar {\psi }}(y)\psi (x).$

Plugging our plane wave expansion for the fermion field into the above equation yields:

$D_{F}(x-y)=\int {\frac {d^{4}p}{(2\pi )^{4}}}{\frac {i(p\!\!\!/+m)}{p^{2}-m^{2}+i\epsilon }}e^{-ip\cdot (x-y)}$

where we have employed the Feynman slash notation. This result makes sense since the factor

${\frac {i(p\!\!\!/+m)}{p^{2}-m^{2}}}$

is just the inverse of the operator acting on ψ(x) in the Dirac equation. Note that the Feynman propagator for the Klein–Gordon field has this same property. Since all reasonable observables (such as energy, charge, particle number, etc.) are built out of an even number of fermion fields, the commutation relation vanishes between any two observables at spacetime points outside the light cone. As we know from elementary quantum mechanics two simultaneously commuting observables can be measured simultaneously. We have therefore correctly implemented Lorentz invariance for the Dirac field, and preserved causality.

More complicated field theories involving interactions (such as Yukawa theory, or quantum electrodynamics) can be analyzed too, by various perturbative and non-perturbative methods.

Dirac fields are an important ingredient of the Standard Model.

See also

• Dirac equation
• Einstein–Maxwell–Dirac equations
• Spin-statistics theorem
• Spinor

References

• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• Peskin, M and Schroeder, D. (1995). An Introduction to Quantum Field Theory, Westview Press. (See pages 35–63.)
• Srednicki, Mark (2007). Quantum Field Theory, Cambridge University Press, ISBN 978-0-521-86449-7.
• Weinberg, Steven (1995). The Quantum Theory of Fields, (3 volumes) Cambridge University Press.