Minkowski spacetime vacuum

The free theory for the quench models is provided by the potential (4), where A = 0 and m2(t) changes signs either instantaneously or for a finite period. In the Minkowski spacetime, we can apply the LvN method simply by letting R = 1. Before the phase transition (rrii = (mg + m2)1/2), all the modes are stable and oscillate around the true vacuum ... 

The P on the left-hand side of Eq. (162) denotes path ordering and the P denotes area ordering [4]. Equation (162) is the result of a round trip or closed loop in Minkowski spacetime with 0(3) covariant derivatives. Equation (161) is a direct result of our basic assumption that the configuration of the vacuum can be described by gauge theory with an internal 0(3) symmetry (Section I). Henceforth, we shall omit the P and P from the left- and right-hand sides, respectively, and give a few illustrative examples of the use of Eq. (162) in interferometry and physical optics. 

The ansatz, upon which these results are based, is that the configuration of the vacuum is described by the doubly connected group 0(3), which supports the Aharonov-Bohm effect in Minkowski spacetime [46]. More generally, the vacuum configuration could be described by an internal gauge space more general than 0(3), such as the Lorentz, Poincare, or Einstein groups. The 0(3)... 

Gauge theory of any symmetry must have two mathematical spaces Minkowski spacetime and the internal gauge space. If electromagnetic theory in the vacuum is a U(l) symmetry gauge field symmetry, there is a scalar internal space of U(l) symmetry in the vacuum. This internal space is the space of the scalar A and A used in the foregoing arguments. In geometric form... 

Gauge theory can be developed systematically for the vacuum on the basis of material presented in Section II. Before doing so, recall that, on the U(l) level, AM exists in Minkowski spacetime and there is a scalar internal gauge space that can be denoted... 

It is also possible to consider the holonomy of the generic A in the vacuum. This is a round trip or closed loop in Minkowski spacetime. The general vector A is transported from point A, where it is denoted Aa 0 around a closed loop with covariant derivatives back to the point Aa () in the vacuum. The result [46] is the field tensor for any gauge group... 

The left-hand side of Eq. (523) denotes around trip or closed loop in Minkowski spacetime [46]. On the U(l) level, this is zero in the vacuum because the line integral... 

But the Minkowski spacetime R4 has trivial cohomology. This means that the Maxwell equation implies that. is a closed 2-form, so it is also an exact form and we can write. = d d, where ( is another potential 1-form in the Minkowski space. Now the dynamical equation becomes another Bianchi identity. This simple idea is a consequence of the electromagnetic duality, which is an exact symmetry in vacuum. In tensor components, with sJ = A dx and ((i = C(1dxt we have b iV = c, /tv — and b iV = SMCV - SvC or, in vector components... 

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