# Quantum Field Theory Representation in Quantum Computation
## 2 Materials and Methods
QFT assumes that spacetime coordinates \(\left(\overset{\rightarrow}{r},t\right)\) are flat [4], meaning that the spacetime curvature caused by gravity is very small, and so the phase contribution by spacetime location \(\left(\overset{\rightarrow}{r}^{\prime},t^{\prime}\right)\) to the time-ordered transition between two quantum field configurations, \(\overset{\rightarrow}{\Psi}_{(\overset{\rightarrow}{r},t)}\) and \(\psi_{(\overset{\rightarrow}{r},t+\Delta t)}\), is for QFT (see Appendix A) approximately:  
\[\frac{\left\langle 0\right|\ \sum_{\text{all configurations}}\left\{\psi_{( \overset{\rightarrow}{r},t+\Delta t)}\overset{\rightarrow}{\Psi}_{( \overset{\rightarrow}{r},t)}\ e^{i\frac{\Delta t}{\hbar}\left[\overset{ \rightarrow}{c}\cdot\Delta t\right]^{3}\mathcal{L}_{int}\left(\overset{ \rightarrow}{r}^{\prime},t^{\prime}\right)}\ \right\}\left|0\right\rangle}{\left\langle 0\right|\ \sum_{\text{all configurations}}\left\{e^{i\frac{ \Delta t}{\hbar}\left[\overset{\rightarrow}{c}\cdot\Delta t\right]^{3} \mathcal{L}_{int}\left(\overset{\rightarrow}{r}^{\prime},t^{\prime}\right)} \right\}\left|0\right\rangle} \tag{3}\]  
A consequence of Equations (2) and (3) making quantum fields compatible with special relativity is that the particles of the electromagnetic field, the photons, are quanta of the \(\left[\overset{\rightarrow}{\phi},-\overset{\rightarrow}{cA}\right]\) quantum field, and the electrons are quanta of the electron quantum field; therefore, both photons and electrons are being created and destroyed, as much as energy-momentum conservation and [PERSON]'s uncertainty rules allow. The breaking of energy-momentum conservation being typically through the occurrence of extremely short-lived virtual particles and virtual antiparticles, which constitute the vacuum state \(\left|0\right\rangle\)[4]. The existence of a quantum electrodynamics dynamic vacuum state \(\left|0\right\rangle_{EM}\) has been experimentally confirmed multiple times and is the theoretical basis for cavity quantum electrodynamics which is one of the major approaches towards QC [7]. The establishment of Equations (2) and (3) is the end of QM and the start of QFT, as several of the tenets of QM need to be abandoned. In QM, the state of a particle is represented in a Hilbert space, and the number of required Hilbert spaces grows with the number of particles, and hence there are as many Hilbert spaces as there are particles. However, in QFT the number of particles is dynamic and often undeterminable, and hence the use of Hilbert spaces becomes completely impractical, except for simplistic models where the number of particles is preserved or is irrelevant. Thus, in QFT the quantum fields are considered to exist everywhere, and it is their interaction that can be used to calculate the probability of certain events. Hence, in QFT, the tensor-product approach cannot be used to assess the occurrence of entanglement of quantum systems and instead the commuting-operator approach is used. In QM, it is often said that a certain electron is mostly located in a certain region and another electron is located in some other region; however, in QFT those two electrons are indistinguishable quanta manifestations of the same electron quantum field with the two quanta associated to different spacetime locations. In QM, there is only one electron quantum field in the Universe and all electrons are quanta of that field. The electron quantum field is a fermionic quantum field because the electron spin is a semi-integer multiple of \(\hbar\), specifically \(\frac{1}{2}\cdot\hbar\), which is referred to as being spin \(\frac{1}{2}\). The Standard Model of Physics (SMP) uses the Feynman perspective of QFT to explain all Physics, except for gravity, and when QFT is assessed, it is usually assumed that it is the SMP that is being considered. The SMP consists of bosonic and fermionic quantum fields.