Why is Quantum Physics Based on Complex Numbers?

Felix Lev

Home/Archive/arXiv

arXiv · hep-th/0309003

Why is Quantum Physics Based on Complex Numbers?

Felix Lev

View original source ↗Read source document ↗Export metadata ↓

Abstract

The modern quantum theory is based on the assumption that quantum states are represented by elements of a complex Hilbert space. It is expected that in future quantum theory the number field will be not postulated but derived from more general principles. We consider the choice of the number field in quantum theory based on a Galois field (GFQT) discussed in our previous publications. Since any Galois field is not algebraically closed, in the general case there is no guarantee that even a Hermitian operator necessarily has eigenvalues. We assume that the symmetry algebra is the Galois field analog of the de Sitter algebra so(1,4) and consider spinless irreducible representations of this algebra. It is shown that the Galois field analog of complex numbers is the minimal extension of the residue field modulo $p$ for which the representations are fully decomposable.

Explore related subjects

hep-th

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Felix Lev. 2006-05-31. Why is Quantum Physics Based on Complex Numbers?. https://arxiv.org/abs/hep-th/0309003

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Holographic Schwinger effect with Translational Symmetry Breaking

We investigate the holographic Schwinger effect in a background with translational symmetry breaking (TSB) at finite chemical potential. The gravitational background is characterized by two independent parameters: the TSB parameter $α$, which controls momentum relaxation, and the chemical potential $μ$, which determines the finite density of the dual field theory. Using the potential analysis method, we derive the total potential governing the pair production process and examine its dependence on $α$, $μ$, the external magnetic field, and the ratio $β=E/E_c$. Our results show that the effects of $α$ and $μ$ on the Schwinger process strongly depend on the dynamical regime. In the subcritical regime, increasing either $α$ or $μ$ lowers the potential barrier and facilitates pair production. However, near and above the critical electric field, the roles of these two parameters become qualitatively different. While increasing the chemical potential lowers the total potential and enhances the Schwinger pair production process, increasing the translational symmetry breaking parameter shifts the potential upward and suppresses the production process. We further show that, at fixed $β=E/E_c$, a perpendicular external magnetic field lowers the effective potential barrier and thereby facilitates the Schwinger process, while the physical electric field changes accordingly through the magnetic-field dependence of $E_c$. The corresponding pair production rate is not independently calculated; instead, its qualitative behavior is characterized through a proxy derived from the total potential. Overall, our analysis provides a comprehensive picture of how translational symmetry breaking, finite density, and external magnetic fields influence holographic non-perturbative pair production.

hep-th↗

arXiv2026

One-loop Corrected Holographic Shear Viscosity to Entropy Density Ratio at Low Temperatures

Near-extremal black holes contain infrared-enhanced quantum fluctuations in their near-horizon near-AdS$_2$ throat, governed in part by Schwarzian modes. We study the effect of these fluctuations on the zero-frequency stress-tensor response of a near-extremal asymptotically AdS$_4$ Reissner--Nordström black brane whose transverse directions are regulated by a finite toroidal quotient. Working directly in four dimensions, we compute the leading correction in the weakly coupled Schwarzian regime $T_q\ll T\ll r_0/L^2$. At one loop, the shear perturbation couples to the $n=2$ would-be zero mode and produces a correction proportional to $T_q/T$. Together with the logarithmic correction to the entropy, this yields a finite-volume correction to $η/s$. Our calculation provides a direct four-dimensional benchmark, complementary to exact two-dimensional treatments, while keeping the asymptotic AdS$_4$ source explicit.

hep-th↗

arXiv2026

Families of Hitchin Systems in Type-D

The Coulomb branch geometry of a 4d $\mathcal{N}=2$ SCFT is encoded in the data of a complex integrable system. In class-S, this is the Hitchin System (of ADE type) on the punctured curves $C$ on which we compactified from 6d to 4d. As we vary the complex structure of $C$, these fit together to form a (nontrivial!) bundle of Hitchin systems over the moduli space of complex structures of $C$ (the ``conformal manifold'' of the family of SCFTs). We carry out that construction for type-D. Compared to the type-A case, the construction is much more complicated because of local constraints at the punctures. Those local constraints were studied in [1]. Here, we work out their implications for the global bundle of spectral (Seiberg-Witten) curves.

hep-th↗