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TheDirac sea is a theoretical model of the electronvacuum as an infinite sea of electrons withnegative energy, now calledpositrons. It was first postulated by the British physicistPaul Dirac in 1930^[1] to explain the anomalous negative-energyquantum states predicted by therelativistically correctDirac equation forelectrons.^[2] The positron, theantimatter counterpart of the electron, was originally conceived of as ahole in the Dirac sea, before its experimental discovery in 1932.^{[nb 1]}

In hole theory, the solutions with negative time evolution factors^{[clarification needed]} are reinterpreted as representing the positron, discovered byCarl Anderson. The interpretation of this result requires a Dirac sea, showing that the Dirac equation is not merely a combination ofspecial relativity andquantum mechanics, but it also implies that the number of particles cannot be conserved.^[3]

Dirac sea theory has been displaced byquantum field theory, though they are mathematically compatible.

Similar ideas onholes in crystals had been developed by Soviet physicistYakov Frenkel in 1926, but there is no indication the concept was discussed with Dirac when the two met in a Soviet physics congress in the summer of 1928.

The origins of the Dirac sea lie in theenergy spectrum of theDirac equation, an extension of theSchrödinger equation consistent withspecial relativity, an equation that Dirac had formulated in 1928. Although this equation was extremely successful in describing electron dynamics, it possesses a rather peculiar feature: for eachquantum state possessing a positive energyE, there is a corresponding state with energy -E. This is not a big difficulty when an isolated electron is considered, because its energy isconserved and negative-energy electrons may be left out. However, difficulties arise when effects of theelectromagnetic field are considered, because a positive-energy electron would be able to shed energy by continuously emittingphotons, a process that could continue without limit as the electron descends into ever lower energy states. However, real electrons clearly do not behave in this way.

Dirac's solution to this was to rely on thePauli exclusion principle. Electrons arefermions, and obey the exclusion principle, which means that no two electrons can share a single energy state within an atom. Dirac hypothesized that what we think of as the "vacuum" is actually the state in whichall the negative-energy states are filled, and none of the positive-energy states. Therefore, if we want to introduce a single electron, we would have to put it in a positive-energy state, as all the negative-energy states are occupied. Furthermore, even if the electron loses energy by emitting photons it would be forbidden from dropping below zero energy.

Dirac further pointed out that a situation might exist in which all the negative-energy states are occupied except one. This "hole" in the sea of negative-energy electrons would respond toelectric fields as though it were a positively charged particle. Initially, Dirac identified this hole as aproton. However,Robert Oppenheimer pointed out that an electron and its hole would be able toannihilate each other, releasing energy on the order of the electron's rest energy in the form of energetic photons; if holes were protons, stableatoms would not exist.^[4]Hermann Weyl also noted that a hole should act as though it has the samemass as an electron, whereas the proton is about two thousand times heavier. The issue was finally resolved in 1932, when thepositron was discovered byCarl Anderson, with all the physical properties predicted for theDirac hole.

Despite its success, the idea of the Dirac sea tends not to strike people as very elegant. The existence of the sea implies an infinite negative electric charge filling all of space. In order to make any sense out of this, one must assume that the "bare vacuum" must have an infinite positive charge density which is exactly cancelled by the Dirac sea. Since the absolute energy density is unobservable—thecosmological constant aside—the infinite energy density of the vacuum does not represent a problem. Only changes in the energy density are observable.Geoffrey Landis also notes^{[citation needed]} that Pauli exclusion does not definitively mean that a filled Dirac sea cannot accept more electrons, since, asHilbert elucidated, a sea of infinite extent can accept new particles even if it is filled. This happens when we have achiral anomaly and a gaugeinstanton.

The development ofquantum field theory (QFT) in the 1930s made it possible to reformulate the Dirac equation in a way that treats the positron as a "real" particle rather than the absence of a particle, and makes the vacuum the state in which no particles exist instead of an infinite sea of particles. This picture recaptures all the valid predictions of the Dirac sea^{[citation needed]}, such as electron-positron annihilation. On the other hand, the field formulation does not eliminate all the difficulties raised by the Dirac sea; in particular the problem of thevacuum possessing infinite energy.

Upon solving the free Dirac equation,

$${\displaystyle i\hslash \frac{\mathrm{\partial }\mathrm{\Psi }}{\mathrm{\partial }t}=(c\hat{\mathit{\alpha }}\cdot \hat{\mathit{p}}+m{c}^2\hat{\beta })\mathrm{\Psi },}$$

one finds^[5]

$${\displaystyle {\mathrm{\Psi }}_{\mathbf{p}\lambda }=N\left(\begin{array}{c}U\\ \frac{(c\hat{\mathit{\sigma }}\cdot \mathit{p})}{m{c}^2+\lambda E_p}U\end{array}\right)\frac{\exp [i(\mathbf{p}\cdot \mathbf{x}-\epsilon t)/\hslash ]}{{\sqrt{2\pi \hslash }}^3},}$$

where

$${\displaystyle \epsilon =\pm E_p,E_p=+c\sqrt{{\mathbf{p}}^2+m^2{c}^2},\lambda =\mathrm{sgn}\epsilon }$$

for plane wave solutions with3-momentump. This is a direct consequence of the relativisticenergy-momentum relation

$${\displaystyle E^2=p^2{c}^2+m^2{c}^4}$$

upon which the Dirac equation is built. The quantityU is a constant2 × 1 column vector andN is a normalization constant. The quantityε is called thetime evolution factor, and its interpretation in similar roles in, for example, theplane wave solutions of theSchrödinger equation, is the energy of the wave (particle). This interpretation is not immediately available here since it may acquire negative values. A similar situation prevails for theKlein–Gordon equation. In that case, theabsolute value ofε can be interpreted as the energy of the wave since in the canonical formalism, waves with negativeε actually havepositive energyE_p.^[6] But this is not the case with the Dirac equation. The energy in the canonical formalism associated with negativeε is–E_p.^[7]

The Dirac sea interpretation and the modern QFT interpretation are related by what may be thought of as a very simpleBogoliubov transformation, an identification between the creation and annihilation operators of two different free field theories.^{[citation needed]} In the modern interpretation, the field operator for a Dirac spinor is a sum of creation operators and annihilation operators, in a schematic notation:

$${\displaystyle \psi (x)=\sum a^{†}(k)e^{ikx}+a(k)e^{-ikx}}$$

An operator with negative frequency lowers the energy of any state by an amount proportional to the frequency, while operators with positive frequency raise the energy of any state.

In the modern interpretation, the positive frequency operators add a positive energy particle, adding to the energy, while the negative frequency operators annihilate a positive energy particle, and lower the energy. For afermionic field, thecreation operator${\displaystyle a^{†}(k)}$ gives zero when the state with momentum k is already filled, while the annihilation operator${\displaystyle a(k)}$ gives zero when the state with momentumk is empty.

But then it is possible to reinterpret the annihilation operator as acreation operator for anegative energy particle. It still lowers the energy of the vacuum, but in this point of view it does so by creating a negative energy object. This reinterpretation only affects the philosophy. To reproduce the rules for when annihilation in the vacuum gives zero, the notion of "empty" and "filled" must be reversed for the negative energy states. Instead of being states with no antiparticle, these are states that are already filled with a negative energy particle.

The price is that there is a nonuniformity in certain expressions, because replacing annihilation with creation adds a constant to the negative energy particle number. Thenumber operator for a Fermi field^[8] is:

$${\displaystyle N=a^{†}a=1-a{a}^{†}}$$

which means that if one replaces N by 1−N fornegative energy states, there is a constant shift in quantities like the energy and the charge density, quantities that count the total number of particles. The infinite constant gives the Dirac sea an infinite energy and charge density. The vacuum charge density should be zero, since the vacuum isLorentz invariant, but this is artificial to arrange in Dirac's picture. The way it is done is by passing to the modern interpretation.

Dirac's idea is more directly applicable tosolid state physics, where thevalence band in asolid can be regarded as a "sea" of electrons. Holes in this sea indeed occur, and are extremely important for understanding the effects ofsemiconductors, though they are never referred to as "positrons". Unlike in particle physics, there is an underlying positive charge—the charge of theionic lattice—that cancels out the electric charge of the sea.

Dirac's original concept of a sea of particles was revived in the theory ofcausal fermion systems, a recent proposal for a unified physical theory. In this approach, the problems of the infinitevacuum energy and infinite charge density of the Dirac sea disappear because these divergences drop out of the physical equations formulated via thecausal action principle.^[9] These equations do not require a preexisting space-time, making it possible to realize the concept that space-time and all structures therein arise as a result of the collective interaction of the sea states with each other and with the additional particles and "holes" in the sea.

• Fermi sea
• Positronium
• Vacuum polarization
• Virtual particle

• Alvarez-Gaume, Luis; Vazquez-Mozo, Miguel A. (2005). "Introductory Lectures on Quantum Field Theory".CERN Yellow Report CERN.1 (96):2010–001.arXiv:hep-th/0510040.Bibcode:2005hep.th...10040A.
• Dirac, P. A. M. (1930)."A Theory of Electrons and Protons".Proc. R. Soc. Lond. A.126 (801):360–365.Bibcode:1930RSPSA.126..360D.doi:10.1098/rspa.1930.0013.JSTOR 95359.
• Dirac, P. A. M. (1931)."Quantized Singularities In The Electromagnetic Fields".Proc. R. Soc. A.133 (821):60–72.Bibcode:1931RSPSA.133...60D.doi:10.1098/rspa.1931.0130.JSTOR 95639.
• Finster, F. (2011). "A formulation of quantum field theory realizing a sea of interacting Dirac particles".Lett. Math. Phys.97 (2):165–183.arXiv:0911.2102.Bibcode:2011LMaPh..97..165F.doi:10.1007/s11005-011-0473-1.ISSN 0377-9017.S2CID 39764396.
• Greiner, W. (2000).Relativistic Quantum Mechanics. Wave Equations (3rd ed.).Springer Verlag.ISBN 978-3-5406-74573. (Chapter 12 is dedicate to hole theory.)
• Sattler, K. D. (2010).Handbook of Nanophysics: Principles and Methods.CRC Press. pp. 10–4.ISBN 978-1-4200-7540-3. Retrieved2011-10-24.

• Quantum field theory
• Vacuum
• Paul Dirac

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