Dirac Negative-Energy Solutions

The negative-energy sector of the Dirac equation has a positive Hilbert norm. It is also not the same as the lower two components of a Dirac-basis spinor. Energy sectors are selected by momentum-dependent projectors; component blocks are selected by a basis. Quantization adds the separate step that interprets the negative-frequency modes through positive-energy antiparticle creation operators.

Required background. Free Dirac Spinors fixes mode labels and normalization; The Dirac Hamiltonian as an Operator constructs the energy projectors. Helpful background. Negative-Energy Solutions discusses sector mixing independently of the Dirac-specific component structure. Use natural units $\hbar=c=1$ below.

Energy sign is not component position

For $m>0$, a positive-energy spinor has Dirac-basis blocks

$$u(p)=\sqrt{E+m} \begin{pmatrix}\chi\\ (\boldsymbol\sigma\cdot\mathbf p)\chi/(E+m) \end{pmatrix},\qquad \chi^\dagger\chi=1.$$

Its upper and lower squared norms are $E+m$ and $E-m$. After dividing by the total $2E$, their fractions are

$$f_{\rm upper}=\frac12\left(1+\frac mE\right),\qquad f_{\rm lower}=\frac12\left(1-\frac mE\right).$$

Nevertheless $P_-u=0$ exactly. The lower block is not an antiparticle probability; it is required by the positive-energy equation at nonzero momentum. In the ultrarelativistic limit the two blocks have nearly equal norms while the state remains entirely in the positive sector. The terminology “large” and “small” components is restricted to a specified low-momentum regime and basis.

Conversely, an upper-only unit spinor $w=(\chi,0)^T$ at fixed nonzero momentum is generally not a positive-energy eigenvector. With $h(\mathbf p)=\boldsymbol\alpha\cdot\mathbf p+\beta m$,

$$\|P_\pm(\mathbf p)w\|^2 =w^\dagger P_\pm w =\frac12\left(1\pm\frac mE\right).$$

At $|\mathbf p|=m$, the negative-sector fraction is $(1-1/\sqrt2)/2\approx0.1464$. A true positive-energy spinor at the same momentum has the same numerical lower-block fraction but zero negative-sector fraction. The two measurements ask different questions.

Norm, spectral energy, and classical electric current

For a negative-frequency mode $v(p)e^{+ip\cdot x}$ with $p^0=E>0$, the differential generators have eigenvalues $-E$ and $-\mathbf p$. Its density is $v^\dagger v=2E>0$, while $\bar vv=-2m$ is a Lorentz scalar bilinear, not the Hilbert norm.

The free first-quantized Hamiltonian is self-adjoint and its evolution is unitary, even though its spectrum has no lower bound. This is not an algebraic inconsistency in the initial-value problem. It becomes an interpretive problem if the entire spectrum is treated as available energies of one isolated electron interacting without further structure. Simply deleting the negative sector with a local component rule does not solve it.

Likewise, a c-number Dirac amplitude carrying parameter $q$ has electric current $q\bar\psi\gamma^\mu\psi$. Inserting a $v$ solution does not automatically change that parameter to $-q$. The antiparticle charge sign comes from the field construction, or from a separately defined charge-conjugate equation, not from the sign of $v^\dagger v$.

What the sea picture was trying to explain

The historical sea picture fills negative-energy electron states and uses the exclusion principle to block further downward transitions. A missing occupied state then has opposite charge and positive energy relative to the filled reference. This gave a useful interpretation of holes, but requires an infinite reference occupation in the continuum and is not the modern fundamental definition of the vacuum.

Contemporary free-field quantization starts with electron and positron operators and a vacuum annihilated by both species’ annihilation operators. It does not require a literal material medium of occupied electrons. The sea account also cannot be transferred unchanged to bosonic scalar particles, for which Pauli blocking is absent.

The operator ordering that supplies the charge sign

Suppress spin and momentum sums and use canonically normalized field modes. The negative-frequency part of the Dirac field carries a creation operator:

$$\widehat\psi\sim u\,a\,e^{-ip\cdot x} +v\,b^\dagger e^{+ip\cdot x}.$$

Fermionic anticommutation gives $bb^\dagger=1-b^\dagger b$. Subtracting the free vacuum constant by normal ordering yields the single-mode contributions

$$:H:=E(a^\dagger a+b^\dagger b),\qquad :Q:=q(a^\dagger a-b^\dagger b).$$

Both excitations have positive energy; their charges are opposite. This is a statement about field operators and the chosen free vacuum. It is not obtained by assigning a negative probability to a c-number mode. Normal ordering here specifies the free reference subtraction, not a universal prescription for every interacting or gravitational vacuum problem.

In a background that does not preserve the chosen particle sector, classical mode mixing remains a useful part of the calculation. To interpret it as production, specify the quantized field, the initial state, and suitable final particle modes. A transition between two components of a four-spinor is not by itself a pair count.

Exercises

1. Compute $\langle\beta\rangle$ for a normalized positive-energy spinor. Can it identify a lower-component antiparticle fraction?

Solution

$\langle\beta\rangle=(E+m-E+m)/(2E)=m/E$. It measures the difference of Dirac-basis block norms, not an energy projector. The state has zero negative-energy weight for every momentum.

2. For the upper-only spinor $w$ above, find the energy expectation and variance at fixed momentum.

Solution

$\langle h\rangle=m$ because the alpha matrices connect opposite blocks. Since $h^2=E^2I$, the variance is $E^2-m^2=\mathbf p^2$. For nonzero momentum it is not an energy eigenstate, despite having only upper components.

3. Explain why the absence of negative Dirac probability does not remove the need for antiparticles in the field description.

Solution

Positive norm establishes a consistent Hilbert-space probability, but does not remove the second spectral sector or ensure fixed particle number under interactions. Quantization reorganizes the modes into positive-energy excitations of opposite charge and supplies states with different particle numbers.

References

• J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw–Hill, 1964 — negative-energy solutions and hole interpretation.
• M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014, doi:10.1017/9781139540940 — fermionic quantization, charge, and energy.
• B. Thaller, The Dirac Equation, Springer, 1992 — spectral sectors and relativistic one-particle interpretation.