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In [[quantum field theory]], a '''fermionic field''' is a [[quantum field]] whose quanta are [[fermion]]s; that is, they obey [[Fermi–Dirac statistics]]. Fermionic fields obey [[canonical anticommutation relation]]s rather than the [[canonical commutation relation]]s of [[bosonic field]]s.
It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>If you liked this post and you would like to get far more details with regards to [http://www.youtube.com/watch?v=90z1mmiwNS8 dentist DC] kindly visit our web site.
 
The most prominent example of a fermionic field is the Dirac field, which describes fermions with [[spin (physics)|spin]]-1/2: [[electron]]s, [[proton]]s, [[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 fermion]]s, 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 [[Double beta decay#Neutrinoless double beta decay|Neutrinoless double-beta decay]] for experimental efforts to determine this).
 
==Basic properties==
Free (non-interacting) fermionic fields obey [[canonical anticommutation relation]]s, i.e., involve the [[anticommutator]]s {''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]],
 
:<math>(i\gamma^{\mu} \partial_{\mu} - m) \psi(x) = 0.\,</math>
 
where γ<sup>μ</sup> are [[gamma matrices]] and ''m'' is the mass. The simplest possible solutions to this equation are plane wave solutions, <math>\psi_{1}(x) = u(p)e^{-ip.x}\,</math> and <math>\psi_{2}(x) = v(p)e^{ip.x}\,</math>. 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,
 
<math>\psi(x) = \int \frac{d^{3}p}{(2\pi)^{3}} \frac{1}{\sqrt{2E_{p}}}\sum_{s} \left(
a^{s}_{\textbf{p}}u^{s}(p)e^{-ip \cdot x}+b^{s \dagger}_{\textbf{p}}v^{s}(p)e^{ip \cdot x}\right).\,</math>
 
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, <math>a^{s}_{\textbf{p}}</math> and <math>b^{s \dagger}_{\textbf{p}}</math> are operators. The properties of these operators can be discerned from the properties of the field. ψ(''x'') and <math>\psi(y)^{\dagger}</math> obey the anticommutation relations
 
:<math>\{\psi_a(\textbf{x}),\psi_b^{\dagger}(\textbf{y})\} = \delta^{(3)}(\textbf{x}-\textbf{y})\delta_{ab},</math>
 
By putting in the expansions for ψ(''x'') and ψ(''y''), the anticommutation relations for the coefficients can be computed.
 
:<math>\{a^{r}_{\textbf{p}},a^{s \dagger}_{\textbf{q}}\} = \{b^{r}_{\textbf{p}},b^{s \dagger}_{\textbf{q}}\}=(2 \pi)^{3} \delta^{3} (\textbf{p}-\textbf{q}) \delta^{rs},\,</math>
 
In a manner analogous to non-relativistic annihilation and creation operators and their commutators, these algebras lead to the physical interpretation that <math>a^{s \dagger}_{\textbf{p}}</math> creates a fermion of momentum '''p''' and spin s, and <math>b^{r \dagger}_{\textbf{q}}</math> 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, <math>\bar{\psi} \ \stackrel{\mathrm{def}}{=}\  \psi^{\dagger} \gamma^{0}</math>, 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 <math>\overline{\psi}\psi\,</math>. This makes the reason for the choice of <math>\bar{\psi} = \psi^{\dagger} \gamma^{0}</math>clear. This is because the general Lorentz transform on ψ is not [[Unitary transformation|unitary]] so the quantity <math>\psi^{\dagger}\psi</math> would not be invariant under such transforms, so the inclusion of <math>\gamma^{0}\,</math> is to correct for this. The other possible non-zero [[Lorentz covariance|Lorentz invariant]] quantity, up to an overall conjugation, constructible from the fermionic fields is <math>\overline{\psi}\gamma^{\mu}\partial_{\mu}\psi</math>.
 
Since linear combinations of these quantities are also Lorentz invariant, this leads naturally to the [[Lagrangian#Lagrangians and Lagrangian densities in field theory|Lagrangian density]] for the Dirac field by the requirement that the [[Euler–Lagrange equation]] of the system recover the Dirac equation.
 
:<math>\mathcal{L}_{D} = \bar{\psi}(i\gamma^{\mu} \partial_{\mu} - m)\psi\,</math>
 
Such an expression has its indices suppressed. When reintroduced the full expression is
 
:<math>\mathcal{L}_{D} = \bar{\psi}_{a}(i\gamma^{\mu}_{ab} \partial_{\mu} - m\mathbb{I}_{ab})\psi_{b}\,</math>
 
Given the expression for ψ(''x'') we can construct the Feynman [[propagator]] for the fermion field:
 
:<math> D_{F}(x-y) = \langle 0| T(\psi(x) \bar{\psi}(y))| 0 \rangle </math>
 
we define the [[time-ordered]] product for fermions with a minus sign due to their anticommuting nature
 
:<math> 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) .</math>
 
Plugging our plane wave expansion for the fermion field into the above equation yields:
 
:<math> 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)}</math>
 
where we have employed the [[Feynman slash]] notation. This result makes sense since the factor
 
:<math>\frac{i(p\!\!\!/ + m)}{p^{2}-m^{2}}</math>
 
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|Einstein–Maxwell–Dirac equations]]
*[[Spin-statistics theorem]]
*[[Spinor]]
 
==References==
*{{cite journal |last=Edwards |first=D. |year=1981 |title=The Mathematical Foundations of Quantum Field Theory: Fermions, Gauge Fields, and Super-symmetry, Part I: Lattice Field Theories |journal=International J. of Theor. Phys. |volume=20 |issue=7 |pages=503–517 |doi=10.1007/BF00669437 |bibcode = 1981IJTP...20..503E }}
* Peskin, M and Schroeder, D. (1995). ''An Introduction to Quantum Field Theory,'' Westview Press. (See pages 35–63.)
* Srednicki, Mark (2007). ''[http://www.physics.ucsb.edu/~mark/qft.html 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.
 
[[Category:Quantum field theory]]
[[Category:Spinors]]

Latest revision as of 13:22, 5 May 2014

It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.

Here are some common dental emergencies:
Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.

At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.

Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.

Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.

Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.

Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.

Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.

In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.

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