Dirac s Hole Theory and the Pauli Principle: Clearing up the Confusion

Transcription

1 Adv. Studies Theor. Phys., Vol. 3, 29, no. 9, Dirac s Hole Theory and the Pauli Princile: Clearing u the Conusion Dan Solomon Rauland-Borg Cororation 82 W. Central Road Mount Prosect, IL 656, USA dan.solomon@rauland.com Abstract In Dirac s hole theory (HT) the vacuum state is generally believed to be the state o minimum energy due to the assumtion that the Pauli Exclusion Princile revents the decay o ositive energy electrons into occuied negative energy states. However recently aers have aeared that claim to show that there exist states with less energy than that o the vacuum[][2][3]. Here we will consider a simle model o HT consisting o ero mass electrons in -D sace-time. It will be shown that or this model there are states with less energy than the HT vacuum state and that the Pauli Princile is obeyed. Thereore the conjecture that the Pauli Princile revents the existence o states with less energy than the vacuum state is not correct. Keywords: Dirac sea - hole theory - vacuum state. Introduction It is well known that there are both ositive and negative energy solutions to the Dirac equation. This creates a roblem in that an electron in a ositive energy state will quickly decay into a negative energy state in the resence o erturbations. This, o course, is not normally observed to occur. This roblem is resumably resolved in Dirac s hole theory (HT) by assuming that all the negative energy states are occuied by a single electron and then evoking the Pauli exclusion rincile to revent the decay o ositive energy electrons into negative energy states. The roosition that the negative energy states are all occuied turns a one electron theory into an -electron theory where. Due to the act that the

2 324 D. Solomon negative energy vacuum electrons obey the same Dirac equation as the ositive energy electrons we have to, in rincile, track the time evolution o an ininite number o states. Also the vacuum electrons in their unerturbed state are unobservable. All we can do is observe the dierences rom the unerturbed vacuum state. It is generally assumed that the HT vacuum state is the state o minimum energy. That is, the energy o all other states must be greater than that o the vacuum state. However a number o aers by the author have shown this is not the case ([][2][3]). It was shown in these aers that states exist in HT that have less energy than the vacuum state. One ossible objection to this result is that it seems to contradict the Pauli exclusion rincile. It is the urose o this aer to show that this is not the case. It will be shown that the existence o states with less energy than the HT vacuum state is erectly consistent with the Pauli rincile. In this aer we will consider a simle quantum theory consisting o non-interacting ero mass electrons in a background classical electric ield. The advantage o such a simle system is that we can easily obtain exact solutions or the Dirac equation or any arbitrary electric otential. This considerably simliies the analysis. In the ollowing discussion we will assume that the system is in some initial state which consists o an ininite number o electrons occuying the negative energy states (the Dirac sea) along with a single ositive energy electron. The system will be then erturbed by an electric ield. Each electron will evolve in time according to the Dirac equation. Ater some eriod o time the electric ield is removed and the change in the energy o each electron can be calculated. The total change in the energy o the system is the sum o these changes. It will be shown that it is ossible to seciy an electric ield so that inal energy is less than the energy o the vacuum state. It will be also shown that this result is entirely consistent with the Pauli exclusion rincile. 2. The Dirac Equation In order to simliy the discussion and avoid unnecessary mathematical details we will assume that the electrons have ero mass and are non-interacting, i.e., they only interact with an external electric otential. Also we will work in - dimensional sace-time where the sace dimension is taken along the -axis and use natural units so that h = c =. In this case the Dirac equation or a single electron in the resence o an external electric otential is, ψ ( t, ) i = Hψ (.) where the Dirac Hamiltonian is given by, H = H qv t (.2),

3 Dirac s hole theory and the Pauli rincile 325 where V, t is an external electrical otential, and q is the electric charge. For ero mass electrons the ree ield Hamiltonian is given as, H = σ i 3 (.3) whereσ 3 is the Pauli matrix with σ 3 =. The solution o (.) can be easily shown to be, ψ t, = W t, ψ t, (.4) H is the Hamiltonian in the absence o interactions, where ( ψ, is the solution to the ree ield equation, ψ ( t, ) i = Hψ (.5) and can be written as, iht ψ t, = e ψ (.6) The quantity W t ) is given by, ic e W = ic (.7) 2 e where c t ) and c2 t ) satisy the ollowing dierential equations, c c = qv (.8) and, c2 c2 = qv (.9) These relationshis also satisy, ( c c2) ( c c2) ( c c2) ( c c2) = 2qV ; = (.) Let ϕ be the eigenunctions o the ree ield Hamiltonian with λ, energy eigenvalue ε ( ) λ,. They satisy the relationshi, H ϕλ, = ελ, ϕλ, (.) where, λ ( ) i ϕλ, ( ) ( ) = e ; ε λ, = λ (.2) 2 L λ and where λ =± is the sign o the energy, is the momentum, and L is the dimensional integration volume. We assume eriodic boundary conditions so that the momentum = 2π rlwhere r is an integer. According to the above

4 326 D. Solomon deinitions the quantities ( ), ( ) ( ) ε = and the quantities ( ), ( ) ε =., ϕ are negative energy states with energy ϕ are ositive energy states with energy, The ϕ orm an orthonormal basis set and satisy, λ, ( ) ( ) ϕλ, ϕλ, d = δλλδ (.3) where integration rom L 2 to L 2 is imlied. I the electric otential is ero then the ϕ evolve in time according to, λ, ( ) iht ( (, ) ) The energy o a normalied wave unction ( t, ) = iλ t λ, λ, λ, ϕ t = e ϕ = e ϕ (.4) ψ is given by, (, ) (, )( (, )) (, ) ξψ t ψ t H V t ψ t d (.5) In the case where V is ero the energy equals the ree ield energy which is given by, ξ ψ t, = ψ t, Hψ t, d (.6) ( ) t = a normalied wave unction ow suose at ψ is seciied and the electric otential is ero. ow aly an electric otential and then remove it at some uture time t. The wave unction has evolved into the state ψ ( t, ) which satisies Eq. (.4). In general the alication o an electric otential will change the ree ield energy o the wave unction. The change in the ree ield energy rom t = to t is shown in the Aendix to be given by, t J Δξ ( t ) = dt V d (.7) where, J t, = qψ t, σ ψ t, (.8) 3 where ψ ( t, ) is given by (.6). The quantity J (, ) current density o the wave unction ψ ( t, ). Recall that ( t, ) time according to the ree ield Dirac equation. Thereore J the ree ield current density. t may be thought o as the ψ evolves in, t wi be called 3. Hole Theory The roosition that the negative energy states are all occuied turns a one electron theory into an -electron theory where. For an -electron theory the wave unction is written as a Slater determinant [], s Ψ (, 2,...,, = ( ) P( ψ(, ψ2( 2, L ψ (, ) (2.)! P

5 Dirac s hole theory and the Pauli rincile 327 where the ( t, ) ψ ( n=, 2, K, ) are a normalied and orthogonal set o wave n unctions that obey the Dirac equation, P is a ermutation oerator acting on the ψ t, and sace coordinates, and s is the number o interchanges in P. ote i a ( t, ) ψ are two wave unctions that obey the Dirac equation then it can be shown b that, ψa ( t, ) ψb( t, ) d= (2.2) ψ t, in (2.) are orthogonal at some initial time then they are Thereore i the orthogonal or all time. n The exectation value o a single article oerator Oo Oe ψ Oo( ) ψ d where ( t, ) is deined as, = (2.3) ψ is a normalied single article wave unction. The -electron oerator is given by, o 2 o n n= (,,..., ) O = O (2.4) which is just the sum o one article oerators. The exectation value o a normalied -electron wave unction is, O (, 2,...,, ) (, 2,..., ) e = Ψ x t Oo Ψ (, 2,...,, dd 2... d (2.5) This can be shown to be equal to, e ψn o n n= (, ) ψ (, ) O = t O t d (2.6) That is, the electron exectation value is just the sum o the single article exectation values associated with each o the individual wave unctions ψ. For examle, the ree ield energy ξ Ψ o the -electron state is, ( Ψ ) = n ( t, ) H n( t, ) d= ( n( t, )) ξ ψ ψ ξ ψ (2.7) n= n= n 4. Time varying electric otential Assume at time t =, the electric otential is ero and the system is in some initial state which is deined in the ollowing discussion. In HT the unerturbed vacuum state is the state where each negative energy wave unction ϕ ( ), is occuied by a single electron and each ositive energy wave unction ϕ ( ), is unoccuied. The energy o the vacuum state is given by summing over the energies o all the negative energy states. The Slater determinant corresonding to this initial vacuum state can be written as, s ( ) ( ) ( ) Ψ (, 2,..., ) = ( ) P( ϕ, ( ) ) ϕ, 2 2 L ϕ, (3.)! P

6 328 D. Solomon where we assume the ollowing ordering; 2 3 K. The total ree ield energy o the unerturbed vacuum state is then, ( ) ( ) E = = vac ε, (3.2) We can add an additional electron rovided it consists o a combination o ( ) ositive energy states ϕ so that it is orthogonal to the vacuum wave unctions ( ) ϕ,,. Let the wave unction that deines this ositive energy electron, at time t =, be given by, ψ ( ), = ϕ (3.3) where the are constant exansion coeicients. Assume that the are selected so that ψ ( ) is normalied. We can write the Slater determinant o this initial state as, s Ψ (,,..., ) = ( ) P! ( ) ( ) ( ) ( ) ( ) ( ψ ϕ ϕ ϕ ) L 2, 2,, 2, P (3.4) Thereore we have, at the initial time t =, a system which consists o the ϕ and a single ositive energy electron unerturbed vacuum electrons, ( ) ( ) ψ. Thereore the total ree ield energy o the system is, ( ) vac ( ) ET = ξ ψ E (3.5) ow we are not really interested in the total energy but in the energy with resect to the unerturbed vacuum state. Thereore we subtract the vacuum ( ) energy E vac rom the above exression to obtain, ( ) ETR, ( ) = ET( ) Evac= ξ( ψ ( ) ) (3.6) which is just the energy o the ositive energy electron. ext, consider the change in the energy due to an interaction with an external electric otential. At the initial time t = the electric otential is ero and the system is in the initial state given by (3.4). ext aly an electric otential and then remove it at some later time t so that, V = or t ; V or < t < t ; V = or t t (3.7) ow what is the change in the energy o the system due to this interaction with the electric otential? Under the action o the electric otential each negative ( ) energy wave unction t. Also the wave unction ( ) ϕ, (,) evolves into the inal state ϕ, (, ) ψ evolves into ψ ( t, ). ote that er (3.7) the electric otential is ero at the initial time t = and the inal time t. Thereore the change in the energy is equal to the change in the ree ield energy. For the negative energy electrons the change in the ree ield energy o each electron is, Δε t = ξ ϕ, t ε (3.8),,,

7 Dirac s hole theory and the Pauli rincile 329 and the change in the energy o the ositive energy electron is, Δξ ( t ) = ξ( ψ t )) ξ( ψ ) ) (3.9) The total change in the energy o the system is then, Δ ET =Δ ξ Δ ε, =Δ ξ ΔEvac (3.) where, ( t ) Δ E = Δ (3.) vac ε, The quantity Δ Evac is the change in energy o the vacuum. Using these results the energy o the system at t with resect to the unerturbed vacuum state is, ETR, t = ETR, Δ ET (3.2) ow we want to evaluate the above quantity. To do this we will use Eq. (.7). For the vacuum electrons the ree ield current density J is given by, ( ) J; t, ) = qϕ, σ3ϕ, (3.3) Reerring to (.2) it is evident that J ; t, ) =. Use this in (.7) to obtain Δε, ( t ) =.. That is, the change in the energy o each o the vacuum electrons is ero. ote that result is indeendent o the alied otential V t ). This yields Δ Evac = Δε, ( t ) = (3.4) r ext we have to determine the change in the energy o the ositive energy electron. The ree ield current density associated with ositive energy electron is, ( J ) t, = qψ t, σ3ψ ( t, ) (3.5) where, ( ) ( ) ψ t, = e ih t ϕ = ϕ t, Use this in (.7) to obtain,,, (3.6) t J ξ ( t ) dt V d (3.7) Δ = ow it is easy to ind a state ( ψ ) ( t, ) so that (, ) J t is non-ero. This can be done by roer selection the exansion coeicients. For examle let, ( ) (, ) i it i i t ψ t = e e e e (3.8) 2L where both and are ositive numbers. In this case, q J = ( cos( ( )( )) (3.9) L

8 33 D. Solomon It is evident that the derivate o this quantity with resect to is non-ero. When J is non-ero it is ossible to ind a V t ) so that Δξ ( t ) is an arbitrarily large negative number. For examle let, J V = g (3.2) where g is a ositive number. Use this in (3.7) to obtain, t J (, ) 2 t Δξ t = g dt d (3.2) ow the integrated quantity is ositive. Thereore as g it is evident that Δξ. Use this in (3.) along with (3.4) to obtain, t J 2 Δ ET = g dt d (3.22) Recall that the energy o the system, with resect to the unerturbed vacuum, at the inal time t is given by E ( t ) = E Δ E. ow due to the act that TR, TR, T Δ E T can be an arbitrarily large negative number then TR, ( ) E t can be negative. Thereore the inal energy o the system can be less than that o the vacuum state. 5. Discussion This result is somewhat surrising. It shows that in HT the unerturbed vacuum state is not the lowest energy state and that it is ossible to extract an unlimited amount o energy rom an initial quantum state. To review the results o the revious sections we started with an initial system consisting o vacuum electrons ( ) in their unerturbed state ϕ, and a ositive energy electron ψ as deined by (3.3). We then aly an electric otential. The result is that each wave unction evolves rom its initial state in accordance with the Dirac equation. We ind that the change in energy o the vacuum electrons rom the initial to inal state is ero. This is true or any electric otential. However when we consider the change in the energy o the wave unction ψ the situation is dierent. In this case i we set u this wave unction so that J is non-ero then we can easily ind an electric otential such that the change in energy o the wave unction ψ can be a negative number with an arbitrarily large magnitude. The net result is that the total energy o the inal system is negative with resect to the energy o the vacuum state. This result is consistent with that o revious work [][2][3]. In the above examle the energy o the vacuum electrons doesn t change and the energy o the wave unction ψ, which was originally ositive, becomes negative. ow wasn t the Pauli rincile suose to revent this? What exactly is the Pauli exclusion rincile? In the context o HT the Pauli rincile is simly the statement that no more than one electron can occuy a given state at given

9 Dirac s hole theory and the Pauli rincile 33 time. Equations (2.) and (2.2) are the mathematical realiation o this statement. The Pauli Princile is a result o the act that i the initial wave unctions in the Slater determinant (see Eq. (2.)) are orthogonal then these wave unctions will be orthogonal or all time. This is a consequence o the act that the individual wave unctions obey the Dirac equation (see Eq. (2.2)). Thereore two electrons cannot end u in the same state. Thereore the calculations erormed in the aer are consistent with the Pauli rincile. All the wave unctions are orthogonal or all time. This means that the conjecture that the Pauli rincile revents the existence o quantum states with less energy than that o the unerturbed vacuum state is not correct. Aendix In this section we will calculate the change in the ree ield energy o a normalied wave unction. Assume at the initial time t = the wave unction is given by ψ ( ). At some uture time t > the wave unction is given by Eq. (.4). The ree ield energy o the state at a given time is given by, ξ t = ψ t, Hψ t, d= ψ tw, HW ψ( t, ) d (.23) Use (.4) and (.3) in the above to obtain, ξ c () t ψ =, t, t d H d c2 ψ ψ ψ (.24) From this we obtain, ( c c2) ( c c2) ξ() t = J ρ d ξ( ) 2q (.25) where J t ) and ρ ( t, ) are the current and charge density, resectively, o the state ψ ( t, ) and are given by, J( t, ) = qψ ( t, ) σ3ψ( t, ) ; ρ( t, ) = qψˆ ( t, ) ψˆ( t, ) (.26) Using the above deinitions along with (.5) and (.3) we can readily show that, J ρ ρ = J ; = (.27) Take the derivative with resect to time o (.25) and use (.27) to obtain, 2 2 ξ() t ρ ( c c2) ( c c2) J ( c c2) ( c c2) = J ρ d 2q (.28) Assume reasonable boundary conditions and integrate by arts to obtain, ξ() t ( c c2) ( c c2) ( c c2) ( c c2) = ρ J d 2q (.29) Use (.) to obtain,

10 332 D. Solomon () ξ t V, t J, t = J d = V d Integrate this rom t = to t to obtain Eq. (.7). (.3) Reerences. D. Solomon. Some dierences between Dirac s hole theory and quantum ield theory, Can. J. Phys., 83 (25), Also arxiv:quanth/ D. Solomon. Some new results concerning the vacuum in Dirac hole theory. Physc. Scr. 74 (26), Also arxiv:quant-h/ D. Solomon. Quantum states with less energy than the vacuum in Dirac hole theory. arxiv:quant-h/7227. Received: February, 29