Positive and negative energies

We can write the Hilbert space of the Dirac equation as a direct sum of the following two orthogonal parts. 

The free Dirac operator acts as a positive operator in Tfpos and as a negative operator in Tfneg- Thus Tfpos (resp. Tfneg) is called the subspace of positive (resp. negative) energies. 

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The functions h (x, p) are the classical relativistic Hamiltonians of spinless particles with positive and negative energies, respectively. 

As noted earlier, for each particle i, there is a discrete spectrum of positive energy bound states and positive and negative energy continuum states. Let us consider a product wave function of the form = i//i(l)i//2(2), a normalizable stationary bound-state eigenfunction of... 

Both FW and ESC methods have been applied to decouple the positive energy states from the negative and mixed states of the two-fermion DBC Hamiltonian [47-49]. Since we are only interested in positive energy solutions, we may start from the original two-fermion DBC Hamiltonian without positive and negative energy projection operators. The two-fermion DBC Hamiltonian for stationary states can be written as... 

Flere, S represents an even operator, that is one that has no matrix elements between positive and negative energy components while is an odd operator having only matrix elements between positive and negative energy components. The conditions for an operator to be even or odd can be expressed more formally an even operator must commute with d [ ,(d] = 0 while an odd operator must anticommute with [S, S]+ =0. 

Note that this relationship is in agreement with our previous equation (3.6) connecting the energy and momentum. Now equation (3.22) reveals the existence of positive and negative energy solutions, i.e. 

In section 3.2 we pointed out that the Dirac Hamiltonian contains operators which connect states of positive and negative energy. What we now seek is a Hamiltonian which is relativistically correct but which operates on the two-component electron functions of positive energy only. We require that this Hamiltonian contain terms representing electromagnetic fields, and Foldy and Wouthuysen [12] showed, by a series of unitary transformations, that such a Hamiltonian can be derived. The Dirac Hamiltonian... 

Bjomtorp, P. (1985). Regional patterns of fat distribution. Ann. Intent. Med. 103,994r-995. Bouchard, C-, and Tremblay, A. (1997). Genetic influences on the response of body fat and fat distribution to positive and negative energy balances in human identical twins. /. Nutt. 127, 943S-947S. 

The positive and negative-energy potentials are plotted in Fig. 2a. The upper potential corresponds to the positive-energy sector of the neutrons in... 

Figiiro 4 The potentials of the positive and negative-energy sector at different potentials become overcritical. 

The sum over n denotes a summation over positive and negative energy states. Evaluating the z-integration by means of complex contour integration yields... 

Figure 6. A superposition of positive and negative energies before and after a Lorentz boost. Apart from the Lorentz contraction, the relative motion between observer and wave packet produces interference effects. The interference is caused by the separation of the negative and positive-energy parts in momentum space.

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