Quantum origin of the Minkowski space

Abstract: We show that D=4 Minkowski space is an emergent concept related to a class of operators in extended Hilbert space with no positive-definite scalar product. We start with the idea of position-like and momentum-like operators (Plewa 2019 J. Phys. A: Math. Theor. 52 375401), introduced discussing a connection between quantum entanglement and geometry predicted by ER=EPR conjecture. We examine eigenequations of the simplest operators and identify D=4 Minkowski space as to be spanned by normalized eigenvectors corresponding to the zero eigenvalue. Both spacetime dimension and signature of the metric are fixed by the regularization procedure. We generalize the result to the case of more general operators, being analogues to quantum fields. We reproduce the Minkowski space again, however, now in a holographic way, as being identified with the conformal boundary of $AdS_5$. We observe an interesting analogy to string theory and, in particular, to AdS/CFT correspondence.