# Quantum Field Theory Representation in Quantum Computation
## 2 Materials and Methods
The QFT is an extension of QM from low-velocity classical relativity to [PERSON]'s special relativity, but if \(m_{0}\) is the rest mass, \(\overset{\rightarrow}{u}=~{}\left(u_{z},u_{y},u_{z}\right)\) is the velocity, \(c\) is the speed of light in vacuum, \(\overset{\rightarrow}{p}=~{}\left(p_{x},p_{y},p_{z}\right)\) is the momentum and \(E\) is the energy, the relation between momentum and the energy is very different between classical relativity where for a free particle \(E=\frac{\left|\overset{\rightarrow}{p}\right|^{2}}{2m_{0}}\) and for special relativity where \(E^{2}=c^{2}\left|\overset{\rightarrow}{p}\right|^{2}+{m_{0}}^{2}c^{4}\). This is important because the sobeposition principle of QM assumes that the relation between energy and momentum must be linear for the energy, which is valid for classic relativity but not special relativity. [PERSON] resolved that problem and obtained a linear relation between energy and momentum for special relativity, but the consequence was that the energy and each of the three components of the momentum were now a four element spinor, respectively \(\overline{E}=~{}\left\{E^{(1)},E^{(2)},E^{(3)},E^{(4)}\right\}\), \(\overset{\rightarrow}{p}=~{}\left\{\overset{\rightarrow}{p}^{(1)},\overset{ \rightarrow}{p}^{(2)},\overset{\rightarrow}{p}^{(3)},\overset{ \rightarrow}{p}^{(4)}\right\}\); so too were the electric potential spinor \(\overset{\rightarrow}{\overline{\phi}}=~{}\left\{\phi^{(1)},\phi^{(2)}, \phi^{(3)},\phi^{(4)}\right\}\) and the vector potential spinor \(\overset{\rightarrow}{A}=~{}\left\{\overset{\rightarrow}{A}^{(1)},\overset {\rightarrow}{A}^{(2)},\overset{\rightarrow}{A}^{(3)},\overset{ \rightarrow}{A}^{(4)}\right\}\) where the electron charge is \(-e\). In the notation developed by [PERSON] usually used in QM, the energy operator is \(i\hbar\frac{d}{dt}\), where \(t\) is time and \(\hbar\) is the reduced Planck constant, and acts on a four-component vector column \(\left\{\psi^{(1)},\psi^{(2)},\psi^{(3)},\psi^{(4)}\right\}\), where each component is a complex function in spacetime. The \(\overline{E}\) spinor notation used here simply represents, in each of the four components, the complex function defined by the action of the energy operator; hence \(\overline{E}=~{}\left\{E^{(1)},E^{(2)},E^{(3)},E^{(4)}\right\}~{}\equiv~{} \left\{i\hbar\frac{d\phi^{(1)}}{dt},i\hbar\frac{d\psi^{(2)}}{dt},i\hbar\frac{d \psi^{(3)}}{dt},i\hbar\frac{d\psi^{(4)}}{dt}\right\}\), and so forth for the other QM operators. For ease of representation, the 4D spinors described above, e.g., \(\overline{E}\), are divided into two 2D spinors, e.g., for the energy spinor \(\overline{E}=~{}\left\{E^{+},E^{-}\right\}\); with the first indicating the particles and the second indicating the opposite of the particles with the meaning of opposite defined below. Thus, \(E^{+}\) 2D spinor indicate particles and \(E^{-}\) indicate the opposite. The physics equations for the QFT are, in this case, the Dirac equation for electromagnetic interactions, which is presented below using our above notation and where the \(2\times 2\) Pauli matrices are represented by \(\sigma_{x}\), \(\sigma_{y}\) and \(\sigma_{z}\):  
\[\left\{ \begin{array}{c}\frac{E^{+}}{c}\\ -\frac{E^{-}}{c}\end{array} \right\}~{}=~{}\left\{ \begin{array}{c}m_{0}c- \frac{c}{c}\phi^{+}+\sigma_{x}\cdot\left(p_{x}^{+}+eA_{x}^{+}\right)~{}+~{} \sigma_{y}\cdot\left(p_{y}^{+}+eA_{y}^{+}\right)~{}+~{}\sigma_{y}\cdot\left(p _{z}^{+}+eA_{z}^{+}\right)\\ m_{0}c+\frac{c}{c}\phi^{-}+\sigma_{x}\cdot\left(p_{x}^{-}+eA_{x}^{-}\right)~{}+~ {}\sigma_{y}\cdot\left(p_{y}^{-}+eA_{y}^{-}\right)~{}+~{}\sigma_{z}\cdot\left( p_{z}^{-}+eA_{z}^{-}\right)\end{array} \right\} \tag{1}\]  
The minus sign in \(E^{-}\) signifies that it behaves differently from \(E^{+}\), but its meaning depends on using either [PERSON]'s or [PERSON]'s perspective. For [PERSON], opposite means the absence of a particle, whereas for [PERSON], opposite means an anti-particle.