An interesting result concerning the lower bound to the energy in the Heisenberg picture

Transcription

1 Aeiron, Vol. 16, No. 2, Aril An interesting result concerning the lower bound to the energy in the Heisenberg icture Dan Solomon Rauland-Borg Cororation 345 W Oakton Skokie, IL 676 USA dan.solomon@rauland.com In quantum theory it is generally assumed that there exists a secial state called the vacuum state and that this state is a lower bound to the energy. However it has recently been demonstrated that this is not necessarily the case for some situations [5]. In order clarify the situation we will consider a very simle field theory in the Heisenberg icture consisting of a quantized fermion field with zero mass articles in 1-1D sace-time interacting with a classical electrical otential. It will be shown that for this examle there is no lower bound to the energy.

2 1. Introduction. Aeiron, Vol. 16, No. 2, Aril In quantum theory it is generally assumed that there exists a secial state called the vacuum state and that this state is a lower bound to the energy. That is, no state can have a value of energy that this less than that of the vacuum state. However it has been shown by the author that this is not necessarily the case for some situations. For examle it can be easily shown that in Dirac s hole theory there exist states with less energy than that of the vacuum state [1][2][3][4]. It has also been recently demonstrated that for quantum field theory in the Heisenberg reresentation there are states with less energy than the vacuum [5]. In order to clarify the situation we examine a very simle field theory in the Heisenberg icture. The field theory will consist of a quantized fermion field consisting of non-interacting fermions with zero mass. This fermion field will interact with a classical otential in 1-1D sace-time. The advantage of this formulation is that it is ossible to obtain exact solutions to the equations of motion. It will be shown that for this field theory there is no lower bound to the energy. In the Heisenberg icture the state vectors Ω are constant in time and the time deendence of the quantum state is carried by the field oerators ψ ˆ ( z, where z is used to reresent the sace dimension. This is in contrast to the Schrödinger icture where the field oerators are constant in time and the time deendence goes with the state vectors. Both ictures are resumed to be equivalent, however this assumtion has been challenged by P.A.M. Dirac [6][7]. Some differences between the two ictures are also discussed [8]. In the rest of this aer we will focus solely on the Heisenberg icture.

3 2. The Heisenberg icture. Aeiron, Vol. 16, No. 2, Aril As was stated in the Introduction we will assume that the electrons have zero mass and are non-interacting, i.e., they only interact with an external electric otential. In addition we will work in 1-1 dimensional sace-time where the sace dimension is taken along the z-axis and use natural units so that = c = 1. This allows us to simlify the discussion and avoid unnecessary mathematical details. In this formulation an exact solution to the equations of motion is readily achieved as will be shown in the following discussion. In the Heisenberg icture the field oerators evolve in time according to the Dirac equation (see Chat. 9 of [9] or Section 8 of [1] or Ref. [5]). For 1-1D sace time the Dirac equation can be written as, ψˆ ( zt, ) i = Hψˆ ( z, (2.1) t where the Dirac Hamiltonian is given by, H = H + qv ( z, (2.2) H is the Hamiltonian in the absence of interactions, V ( z, t ) where is an external electrical otential, and q is the electric charge. For zero mass electrons the free field Hamiltonian is given as, H = σ i 3 (2.3) 1 whereσ 3 is the Pauli matrix with σ 3 = 1. If the electrical otential is zero then the energy of a normalized state vector Ω is given by,

4 Aeiron, Vol. 16, No. 2, Aril ˆ ˆ ξ t = Ω ψ z, t Hψ z, t dz Ω εr (2.4) where ε R is a renormalization constant which is normally defined so that the energy of the vacuum state is zero. Since this is the energy when the electric otential is zero we will sometimes refer to it as the free field energy. The question we want to address is whether or not there is a lower bound to the free field energy. The way we will determine this is as follows. At the initial time t = the electric otential is zero and the system is assumed to be in the initial unerturbed state. In this initial unerturbed state the field oerator is defined by ψˆ( z,) = ψˆ ( z), where ψ ˆ ( z) is discussed in the Aendix, and the state vector is given by Ω. The initial energy is ξ. Next aly an electric otential and then remove it at some later time t f so that, V = for t ; V for < t < tf ; V = for t tf (2.5) The field oerator ˆ ( z, (2.1) with the initial condition ψ ( z,) ψ ( z) ψ will evolve in time according to Eq. ˆ ˆ =. This will, in general, result in a change in the energy. At t f, where the alied otential has been set back to zero, the energy is given by ξ ( t f ). Therefore the change in energy from t =, to the final time, t f is given by, Δξ t = ξ t ξ (2.6) ( f ) ( f )

5 Aeiron, Vol. 16, No. 2, Aril It will be shown that ξ ( t f ) Δ can be a negative number with an arbitrarily large magnitude. Therefore, there is no lower bound to ξ t. In order to calculate ξ ( t f ) f (2.4) to obtain, Δ take the time derivative of ( z ψˆ, dξ () t t dt Hψˆ ( z, = Ω dz Ω (2.7) ˆ ψ ( zt, ) + ψˆ ( zt, ) H t Use (2.1) along with (2.2) and (2.3) to obtain, ψˆ ( zt, ) ψˆ( zt, ) i ψˆ ( zt, ) qv () ( zt, ) ψˆ ( zt, ) σ3 dξ t + = Ω dz dt Ω (2.8) 2 ˆ ψ ( zt, ) iψ ˆ ( z, 2 + qψˆ ( zt, ) σ3 ( V( zt, ) ψˆ( zt, )) Integrate by arts to obtain, dξ ( ( ˆ ˆ = q V zt, Ω ψ zt, σ3ψ ( zt, )) Ω dz (2.9) dt Now, in order to continue this analysis, we need to solve the Dirac equation (2.1). The solution of (2.1) can be easily shown to be,

6 where ˆ ( z, Aeiron, Vol. 16, No. 2, Aril ψ ( z, W ( z, ψ ( z, ˆ ˆ = (2.1) ψ is the solution to the free field equation, ( z ψˆ, i = H ˆ ψ ( z, (2.11) t and can formally be written as, iht ψˆ z, t = e ψˆ z (2.12) The quantity W ( z, t ) is given by, ic1 e W( z, = ic (2.13) 2 e 1, 2 z, t satisfy the following differential equations, c1 c1 + = qv (2.14) t and, c2 c2 = qv (2.15) t Use (2.1) in (2.9) to obtain, where c ( z t ) and c ( dξ (, ) ( ˆ (, ) (, ) 3 (, ) ˆ = q V zt Ω ψ zt W zt σ W ztψ ( zt, )) Ω dz dt (2.16) W z, t σ W z, t = σ in the above to obtain, Use 3 3

7 where, Aeiron, Vol. 16, No. 2, Aril dξ t J z, t = V( z, dz dt (2.17) (, ) ψˆ (, ) σ ψˆ (, ) J z t = q Ω z t z t Ω (2.18) 3 Integrate this from t = to t f to obtain, t f, ( z, ( tf ) dt V ( z dz (2.19) Δξ = J Therefore at time t f the free field energy is given by, ( tf ) ( tf ) ξ =Δξ + ξ (2.2) Note that in (2.19) the quantity J ( z, z is indeendent of V ( z, t ). This is evident from (2.18) and (2.12). Assume for the moment that J ( z, z is non-zero. If this is the case then it is easy to show that we can always find a V ( z, t ) which makes ( t ) Δξ f a negative number with an arbitrarily large magnitude. For examle let, ( z J, V ( z, = f for < t < tf where f is a constant. Use this in (2.19) to obtain, ( f ) Δξ = ( z, 2 t f J t f dt dz (2.21) (2.22)

8 Aeiron, Vol. 16, No. 2, Aril Now as f + it is evident that ξ ( t f ) Δ. This means that an arbitrarily large amount of energy has been extracted from the quantum state due to its interaction with the electric otential ξ t. and that there is no lower bound to the final energy ( f ) 3. Discussion. This result may seem somewhat surrising because it contradicts the widely held assumtion that there is a lower bound to the energy in quantum field theory. Therefore is worth carefully reviewing the assumtions that lead to these results. The first and main assumtion is that the field oerators obey the Dirac equation. Other than this we aly the normal rules of algebra and calculus to obtain (2.19). Note that we don t even use the commutation algebra so issues involving anomalous commutators are not a factor in these results. Now to obtain the final result we assume that the quantum state has been set u so that J ( z, z is non-zero. Now what is J ( z, t )? J ( z, t ) is the free field current exectation value of the normalized state vector Ω. That is, it is the current of the system in the absence of an electric otential. This is why it is indeendent of V ( z, t ). Basically then we are assuming that a state vector Ω exists where J ( z, z is non-zero. Now how do we know that this is the case? It can be easily shown that when the field oerator is exanded in the usual manner in terms of creation and annihilation oerators that there are states that satisfy this condition. This is shown in the Aendix. Therefore we have the following conclusion regarding our very simle field theory - if the field oerator obeys the Dirac

9 Aeiron, Vol. 16, No. 2, Aril equation (2.1) and a state exists for which, J z t z is non-zero then there is no lower bound to the free field energy. This result is consistent with references [1-4] which show that there exist states with less energy than the vacuum in Dirac s hole theory and Ref. [5] in which it was shown that there exist states with less energy than the vacuum for quantum field theory in the Heisenberg icture. Aendix In the main body of this article we have not exressed the field oerators in terms of creation and destruction oerators because this was not necessary to achieve the main results. We will show that when this is done it is easy to shown that states exist where J ( z, z is non-zero. Recall that the field oerator ψ ˆ ( z, must satisfy Eq. (2.11). In addition, it must satisfy the usual equal time anti-commutation relationshi, ψˆ z, t, ψˆ z, t = δ δ z z (A.1) {, α, β } αβ Based on this we can write the field oerator as, ˆ ˆ ψˆ zt, = bϕ zt, + dϕ zt, ( ( ) ) 1, 1, ( ) ( ) ( ) ( z = bˆ ( z + dˆ ( z ψˆ, ϕ, ϕ, 1, 1, r (A.2) where the b ˆ ( b ˆ ) are the destruction(creation) oerators for an electron associated with the state ( ϕ ) 1, ( z, and the ˆ d ( d ˆ ) are the destruction(creation) oerators for a ositron associated with the z t. They satisfy the anticommutator relationshis, ϕ 1,, state { dˆ, } ˆ dq = δ q; { ˆ, } ˆ b b = δ (A.3) q q

10 Aeiron, Vol. 16, No. 2, Aril 29 2 where all other anti-commutators are zero. The vacuum state is defined by, ˆ ˆ d = b = and dˆ = bˆ = for all (A.4) Let ( ) λ, z ϕ be the eigenfunctions of the free field Hamiltonian with energy eigenvalue ε where, ϕ ( ) ( z) λ, λ,. They satisfy the relationshi, ( ) ( ) λ, λ, λ, H ϕ z = ε ϕ z (A.5) λ 1+ 1 = e 2 L λ 1 iz ; ε λ, λ = (A.6) and where λ =± 1 is the sign of the energy, is the momentum, and L is the 1 dimensional integration volume. We assume eriodic boundary conditions so that the momentum = 2π r L where r is an integer. According to the above definitions the quantities ϕ 1, ( z) are negative energy states with energy ε 1, = and the quantities 1, ( z) ( The ) ϕ + are ositive energy states with energy ε + =. λ, z 1, ϕ form an orthonormal basis set and satisfy, ( ) ϕλ, z ϕλ, z dz = δλλδ (A.7) where integration from L 2 to + L 2 is imlied. If the electric otential is zero then the ϕ evolve in time according to, λ, z

11 Aeiron, Vol. 16, No. 2, Aril ( ) iht ( ) iλ t ( ) λ, z, t = e λ, z = e λ, ( z) (A.8) ϕ ϕ ϕ Now consider the state vector, 1 Ω = ( bˆ ˆ + b ) 2,1 q (A.9),1 where both and q are ositive numbers. Referring to the definitions in Section II we can show that the current exectation value of this state is, 1 J ( z, = ( 1+ cos( ( q)( z )) (A.1) 2L It is obvious, then, that J z t z is non-zero.,

12 Aeiron, Vol. 16, No. 2, Aril References [1] D. Solomon. Some differences between Dirac s hole theory and quantum field theory, Can. J. Phys., 83, 257, (25). arxiv:quant-h/ [2] D. Solomon. Some new results concerning the vacuum in Dirac hole theory. Physc. Scr. 74 (26) arxiv:quant-h/6737. [3] D. Solomon. Quantum states with less energy than the vacuum in Dirac hole theory. arxiv:quant-h/ [4] D. Solomon Dirac s hole theory and the Pauli rincile: clearing u the confusion. aixiv: [5] D. Solomon, Some new results concerning the QFT vacuum in the Heisenberg icture, Aerion, 15, No. 2, Aril 28 (567). [6] P.A.M. Dirac, Quantum mechanics without the deadwood, Phys. Rev. 119 (1965), B684. [7] P.A.M. Dirac, Foundations of quantum mechanics, Nature, No (July 11, 1964), 115. [8] D. Solomon The Heisenberg versus Schrödinger icture and the roblem of gauge invariance. aixiv: [9] W. Greiner, B. Muller, and J. Rafelski. Quantum electrodynamics of strong fields. Sringer-Velag, Berlin, [1] W. Pauli. Pauli Lectures on Physics Vol 6 Selected Toics in Field Quantization, MIT Press, Cambridge, Massachusetts, 1973.