Thus, the falling elevator can only be used to define a reference frame within an infinitesimally small region where the gravitational field can be considered uniform.
Over a finite region, the variation of the field may be sufficiently large for the acceleration of the particle not to be completely canceled. We see that an important difference between Special Relativity and GR is that a reference frame can only be defined locally or at a single point in a gravitational field.
This creates a fundamental problem. To illustrate this difficult point, let us now suppose that there are many more physicists in nearby falling elevators. Each physicist makes an independent measurement so that the path of particle in the gravitational field can be determined. The measurements are made in separate elevators at different locations in the field.
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Clearly, one cannot perform an ordinary Lorentz transformation between the elevators. If the different elevators were related only by Lorentz transformation, the acceleration would have to be independent of position and the gravitational field could not decrease with distance from the source. Einstein solved the problem of relating nearby falling frames by defining a new mathematical relation known as connection.
To understand the meaning of a connection, let us consider a four-vector A which represents some physically measured quantity. Now suppose that the physicist in the elevator located at x observes that A changes by an amount dA and a second physicist in a different elevator at x observes a change in dA.
In Special Relativity, the differential dA is also a vector like A itself. The relation 2. We can no longer assume that the transformation from x to x is linear in GR. Thus, we must write for dA the general expression. Such terms are actually quite familiar in physics. They occur in curvilinear coordinate systems. These curvilinear coefficients are denoted by the symbol. They are also called affine connections or Christoffel symbols .
It is important to note that the gravitational connection is not simply the result of using a curvilinear coordinate system. The value of the connection at each point in spacetime is dependent on the properties of the gravitational field.
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The field is important in the determination of the relative orientation of the different falling elevators in the same way that the upward direction on the surface of the earth varies from one position to another. The analogy with curvilinear coordinate systems merely indicates that the mathematical descriptions of free-falling frames and curvilinear coordinates are similar. Einstein generalized this similarity and arrived at the revolutionary idea of replacing gravity by the curvature of spacetime . Let us briefly summarize the essential characteristics of GR that Weyl would have utilized for his new gauge theory.
First of all, GR involves a specific force, gravitation, which is not studied in Special Relativity.
However, by studying the properties of coordinates frames just as in Special Relativity, one learns that only local coordinates can be defined in a gravitational field. This local property is required by the physical behavior of the field and leads naturally to the idea of a connection between local coordinate frames.
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Thus the essential difference between Special Relativity and GR is that the former is a global theory while the latter is a local theory. This local property was the key to Weyls gauge theory . In Section 3 we will develop extensively this idea. Weyl went a step beyond GR and asked the following question: if the effects of a gravitational field can be described by a connection which gives the relative orientation between local frame in spacetime, can other forces of nature such as Electromagnetism also be associated with similar connections? Generalizing the idea that all physical measurements are relative, Weyl proposed that absolute magnitude or norm of a physical vector also should not be an absolute quantity but should depend on its location in spacetime.
A new connection would then be necessary in order to relate the lengths of vectors at different positions. This is the scale or gauge invariance. It is important to note here that the true significance of Weyls proposal lies on the local property of gauge symmetry and not in a special choice of the norm or gauge as a physical variable. The assumption of locality is a powerful condition that determines not only the general structure but many of the detailed features of gauge theory. Weyls gauge invariance can be easily expressed in mathematical form .
Let us consider a vector at position x with norm given by f x. The factor S x is defined for convenience to equal unity at the position x. The derivative S is the new mathematical connection associated with the gauge change. Weyl identified the gauge connection S with the electromagnetic potential A. Since the forms of 2. Unfortunately, it was soon pointed out that the basic idea of scale invariance itself would lead to conflict with known physical facts . Some years later, Bergmann noted, that Weyls original interpretation of gauge invariance would also be in conflict with Quantum Theory.
Since the wavelength is determined by the particle mass M, it cannot depend on position and thus contradicts Weyls original assumption about scale invariance. Despite the initial failure of Weyls gauge theory, the idea of a local gauge symmetry survived. It was well known that Maxwells equations were invariant under a gauge change. However, without an acceptable interpretation, gauge invariance was regarded as only an accidental symmetry of Electromagnetism. The gauge transformation property in Eq. Only the electric and magnetic fields were considered to be real and observable.
Gauge symmetry was retained largely because it was useful for calculations in both classical and quantum electrodynamics. In fact a lot of problems in electrodynamics can often be most easily solved by first choosing a suitable gauge, such as the Coulomb gauge or Lorentz gauge, in order to make the equations more tractable . It is clear that the electromagnetic interaction of charged particles could be interpreted as a local gauge theory within the context of Quantum Mechanics.
In analogy with Weyls first theory, the phase of a particle wavefunction can be identified as a new physical degree of freedom which is dependent on the spacetime position. The value can be changed arbitrarily by performing purely mathematical phase transitions on the wavefunction at each point. Therefore, there must be some connections between phase value nearby points.
The role of this connection is payed by the electromagnetic potential.
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This strict relation between potential and the change in phase is clearly demonstrated by the AharonovBohm effect . Thus by using the phase of wavefunction as the local variable instead of the norm of a vector, Electromagnetism can be interpreted as a local gauge theory very much as Weyl envisioned. Gauge transformations can be viewed as merely phase changes so that they look more like a property of Quantum Mechanics than Electromagnetism.
In addition, the symmetry defined by the gauge transformations does not appear to be natural. The set of all gauge transformations forms a one-dimensional unitary group known as the U 1 group. This group does not arise from any form of coordinate transformation like the more familiar spin-rotation group SU 2 or Lorentz group. Thus, one has lost the original interpretation proposed by Weyl of a new spacetime symmetry. The status of gauge theory was also influenced by the historical fact that Maxwell had formulated Electromagnetism long before Weyl proposed the idea of gauge invariance.
Therefore, unlike the GR, the gauge symmetry group did not play any essential role in defining the dynamical content of Electromagnetism. This sequence of events was to be completely reversed in the development of modern gauge theory [,,]. In , Yang and Mills proposed that the strong nuclear interaction can be described by a field theory like Electromagnetism.
They postulated that the local gauge group was the SU 2 isotopic-spin group. This idea was revolutionary because it changed the very concept of the identity of an elementary particle. If the nuclear interaction is a local gauge theory like Electromagnetism, then there is a potential conflict with the notion of how a particle state.
For example, let us assume that we can turn off the electromagnetic interaction so that we cannot distinguish the proton and neutron by electric charge. We also ignore the small mass difference. We must then label the proton as the up state of isotopic spin 12 and the neutron as the down state. But if isotopic spin invariance is an independent symmetry at each point in spacetime, we cannot assume that the up state is the same at any other point. The local isotopic spin symmetry allows to choose arbitrarily which direction is up at each point without reference to any other point.
Given that the labeling of a proton or a neutron is arbitrary at each point, once the choice has been made at one location, it is clear that some rule is then needed in order to make a comparison with the choice at any other position. The required rule, as Weyl proposed originally, is supplied by a connection. A new isotopic spin potential field was therefore postulated by Yang and Mills in analogy with the electromagnetic potential. However, the greater complexity of the SU 2 isotopic-spin group as compared to the U 1 phase group means that YangMills potential will be quite different from the electromagnetic field.
In Electromagnetism, the potential provides a connection between the phase values of the wavefunction at different positions. In the YangMills theory, the phase is replaced by a more complicated local variable that specifies the direction of the isotopic spin. An obvious way to relate the up states at different locations x and y is to ask how much the up state at x needs to be rotated so that it is oriented in the same direction as the up state at y. This suggests that the connection between isotopic spin states at different points must act like isotopic spin rotation itself.
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In other words, if a test particle in the up state at x is moved through the potential field to position y, its isotopic spin direction must be rotated by the field so that it is pointing in the up direction corresponding to the position y. We can immediately generalize this idea to states of arbitrary isotopic spin.
Since the components of an isotopic spin state can be transformed into another one by elements of the SU 2 group, we can conclude that the connection must be capable of performing the same isotopic spin transformations as the SU 2 group itself. This idea that the isotopic spin connection, and therefore the potential, acts like the SU 2 symmetry group is the most important result of the YangMills theory.
This concept lies at the heart of the local gauge theory. It shows explicitly how the gauge symmetry group is built into the dynamics of the interaction between particles and fields.