Abstract
Relativistic invariance of the vacuum is (or follows from) one of the Wightman axioms which is commonly believed to be true. Without these axioms, here we present a direct and general proof of continuous relativistic invariance of all real-time vacuum correlations of fields, not only scattering (forward in time), based on closed time path formalism. The only assumptions are basic principles of relativistic quantum field theories: the relativistic invariance of the Lagrangian, of the form including known interactions (electromagnetic, weak and strong), and standard rules of quantization. The proof is in principle perturbative leaving a possibility of spontaneous violation of invariance. Time symmetry is however manifestly violated.
Highlights
In this paper, we confirm the common intuition by a direct proof, not relying on Wightman axioms
We show that zero-temperature vacuum correlations of fields at real spacetime points are invariant under continuous transformations of a reference frame
We shall use the framework of the closed time path formalism (CTP) [14,15,16] where correlations are defined on the complex path
Summary
We confirm the common intuition by a direct proof, not relying on Wightman axioms. We show that zero-temperature vacuum correlations of fields at real spacetime points are invariant under continuous transformations of a reference frame. We shall use the framework of the closed time path formalism (CTP) [14,15,16] where correlations are defined on the complex path (going downwards with respect to imaginary part). It is defined in a particular reference frame, the invariance is not manifest. Even if the dynamics is symmetric with respect to time reversal, CTP is not (except special cases, e.g. at space-like points or in scattering problems).
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