Minkowski spacetime

Much more exciting is the E representation, which requires 2x2 matrices  

We have usually referred to the operations of a group generically as multiplications. As mentioned earlier, addition can also be considered a group operation. For example, here is a cute matrix representation which mirrors the addition of two numbers x + y  

Suppose that at t = 0, a light flashes at the origin, creating a spherical wave propagating outward at the speed of light c. The locus of the wave front will 

According to Einstein s special theory of relativity, the wave will retain its spherical appearance to every observer, even one moving at a significant fraction of the speed of light. This can be expressed mathematically as the invariance of the differential element 

This contrived Euclidean geometry does not change the reality that time is fundamentally very different from a spatial variable. It is current practice to accept 

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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 ... 

In order to understand interferometry at a fundamental level in gauge field theory, the starting point must be the non-Abelian Stokes theorem [4]. The theorem is generated by a round trip or closed loop in Minkowski spacetime using covariant derivatives, and in its most general form is given [17] by... 

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. 

This result, however, is an identity of Minkowski spacetime itself, namely, 8 8 operating on a function of produces the same result as 8V8M operating on a function of . Equation (879) does not mean that Aa can take any value. We reach the important conclusion that the vector identity (872) of U(l) is a property of three-dimensional space itself and can always be interpreted as such. Therefore even on the U(l) level, Eq. (872) does not mean that % can take any value. Even on the U(l) level, therefore, potentials can be interpreted physically, as was the intent of Faraday and Maxwell. On the 0(3) level, potentials are always physical. 

The classical Yang-Mills equations of SU(2) gauge theory in the Minkowski spacetime R1,3 form the system of nonlinear second-order partial differential equations of the form... 

There are several major implications of the Jacobi identity (40), so it is helpful to give some background for its derivation. On the U(l) level, consider the following field tensors in c = 1 units and contravariant covariant notation in Minkowski spacetime ... 

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)... 

The method of functional variation in Minkowski spacetime is illustrated hrst through the Lagrangian (in the usual reduced units [46])... 

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... 

The Sagnac effect is therefore due to a gauge transformation and a closed loop in Minkowski spacetime with 0(3) covariant derivatives. 

Physical optics, and interferometry in general, are described by the phase equation of 0(3) electrodynamics, Eq. (524). The round trip or closed loop in Minkowski spacetime is illustrated as follows ... 

The received view, in which the phase factor of optics and electrodynamics is given by Eq. (554), can describe neither the Sagnac nor the Tomita-Chiao effects, which, as we have argued, are the same effects, differing only by geometry. Both are non-Abelian, and both depend on a round trip in Minkowski spacetime using 0(3) covariant derivatives. 

Equation (482) is a simple form of the non-Abelian Stokes theorem, a form that is derived by a round trip in Minkowski spacetime [46]. It has been adapted directly for the 0(3) invariant phase factor as in Eq. (547), which gives a simple and accurate description of the Sagnac effect [44], A U(l) invariant electrodynamics has failed to describe the Sagnac effect for nearly 90 years, and kinematic explanations are also unsatisfactory [50], In an 0(3) or SU(2) invariant electrodynamics, the Sagnac effect is simply a round trip in Minkowski space-time and an effect of special relativity and gauge theory, the most successful theory of the late twentieth century. There are open questions in special relativity [108], but no theory has yet evolved to replace it. 

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... 

In special relativity we deal not just with space, but with flat four-dimensional spacetime, Minkowski spacetime, with metric... 

If we replace the space part here by expression (5), we still deal the Minkowski spacetime. In fact... 

The corresponding geometrical structure is known as Minkowski spacetime. 

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