C_Quigley(1) on gauge origin

On the Origins of Gauge Theory Callum Quigley April 14, 2003 1 Introduction We know the Universe to be governed by four fundamental interactions: namely, the strong and the weak nuclear forces, electromagnetism and grav- itation. It is a driving concept to unify these forces into a single, compre- hensive theory. Though this task is far from its completion, there has been much progress. The first great landmark in its development is attributable to James Maxwell, who in 1864 brought together the seemingly unrelated concepts of electricity, magnetism and optics into the now well known theory of electro- magnetism. Nearly a hundred years later, the weak force too, was combined with the electromagnetic by Glashow, Weinberg and Salam, giving rise to the electroweak theory. Currently, attempts are being made to find a Grand Unified Theory which would explain all the forces, except gravity, as man- ifestations of the same fundamental interaction. We believe such a theory is plausible because these forces are all governed by the same principle: the gauge principle. In fact, we shall see that gravity also obeys this rule, which begs the question, ”Are the four known forces all aspects of some single unified force?” Well, nobody knows, nor is it the purpose of this paper to chase that dream. Rather, we will investigate some of the historical devel- opments which transformed this fundamental notion from a triviality into a cornerstone of physics. Between the times of Maxwell and Salam, there were a number of other advances in unification. Many theories, such as Hermann Weyl’s attempt to unify gravity and electricity in 1918, had to be abandoned. However, in Weyl’s case, we shall see that a slight modification of his original proposal 1 forms the foundation of what is now known as gauge theory. The generaliza- tion of this concept, discovered by Yang and Mills, is the framework which explains both nuclear forces. We will restrict our attention, to the gravita- tional and electromagnetic (EM) forces. In particular, how they are both derivable from the gauge principle. 2 First Notions of Gauge Invariance Roughly speaking, the gauge principle states: If a physical system is invariant with respect to some global (space- time independent) group of continuous transformations, G, then it remains invariant when that group is considered locally (space- time dependant), that is G→G(x). Although this formulation is incomplete, we shall see under what con- ditions local invariance is possible. Throughout the paper, we’ll use the following notation. Notation Indices run from 0 to 3, unless otherwise noted. Square brackets [ ] denote anti-symmetrization. Repeated indices are summed over. Tensor indices are raised and lowered in the usual manner by the metric. Finally, we setc= 1. 2.1 Electromagnetism The gauge principle was first recognized in electromagnetism, but in a rather trivial sense. We require the following definitions: Definition 1a). The electromagnetic 4-current density jµ={ρ,j}, is a 4- vector where ρis the electric charge density and jis the 3-dimensional electric current. b)The electromagnetic 4-potential Aµ(x) ={φ(x),A(x)}, is 1-form where φ andAare the electric and magnetic potentials, respectively. c)Finally, the electromagnetic field tensor Fµν=∂µAν−∂νAµ, is a 2-form which encodes the EM information, via Maxwell’s equations. 2 With this notation, and appropriate units, Maxwell’s equations are com- pactly written: ∂[αFµν]= 0and ∂µFµν=jν. (1) Because of the equality of mixed partials, it follows that the transforma- tion Aµ(x)→A/prime µ(x) =Aµ(x) +∂µα(x), (2) leavesFµνunchanged, for any differentiable scalar function α(x). Thus, Maxwell’s equations are unaltered by adding a gradient. Such a transforma- tion is now termed a gauge transformation , for reasons that will become more clear. Furthermore, for fixed xit is easy to show that the transformations (2) form a commutative (abelian) group, with a single continuous parameter α(x). For a long time, the EM potential Aµwas thought of only a mathematical tool for simplifying calculations, only the field Fµνhad any physical reality. So this gauge freedom, that is the ability to add a gradient onto the potential, was originally considered useful, but unphysical. 2.2 General Relativity The idea of gauge invariance was first appreciated in Einstein’s theory of General Relativity (GR). In fact, GR is derivable from the gauge principle, where the gauge transformations are rigid motions in spacetime. To see this, we must introduce the Christoffel connection, Γλ µν=1 2gλσ(∂νgµσ+∂µgνσ−∂σgµν) where thegµνare the spacetime metric coefficients. An infinitesimal space- time interval’s squared length is given by the 2-form ds2=gµνdxµdxν. (3) When a (co)vector is parallel transported an amount dxν, its components vary as dvλ=−vµΓλ µνdxνor dvµ=vλΓλ µνdxν. (4) 3 Since we write the scalar product between two vectors (at the same point) as u·v=gµνuµvν, then a vector’s squared length is given by |v|2= (gµνvµvν) = (vνvν). We see that this length is invariant under parallel transport dv2=d(vνvν) (5) =dvνvν+vνdvν =vλΓλ µνdxµvν−vνvµΓν µλdxλ = 0. In order for derivatives to remain co-ordinate invariant, that is gauge in- variant, we must modify the partial derivative operator. Otherwise, a change of co-ordinates, from primed to unprimed, for a covector’s partial derivative yields: ∂µ/primevν/prime= (∂µ/primexµ∂νxν/prime∂µ+∂µ/primexµ∂µνvν/prime)vν. (6) The first term is fine, however, the second term is not a tensorial transfor- mation. So, instead we use the covariant derivative, ∇µ(Γ), defined as (∇µ)λ ν=δλ ν∂µ+ Γλ µν (whereδα βis the Kronecker delta) so that ∇µ/primevν/prime=∂µ/primexµ∂νxν/prime∇µvν. (7) In other words, the covariant derivative transforms tensorialy. We will see that covariant derivatives are at the heart of gauge theory; through them, global invariance is preserved locally. The final essential geometric ingredient for GR is the Riemann curvature tensor, which can be expressed in terms of the connection, or the covariant derivative, as Rλ σµν =∂µΓλ σν−∂νΓλ σµ+ Γλ αµΓα σν−Γλ ανΓα σµ = [∇σ,∇µ]λ ν. Note that the second definition highlights the non-commutivity of parallel transport, which tells us about the curvature of spacetime. We will write the contracted Riemann tensor Rλ µνλ≡Rµν, and the Ricci scalar R≡Rν ν. Now we can write Einstein’s field equations for gravitational interactions: Rµν−1 2Rgµν=Tµν (8) whereTµνis the (symmetric) stress-energy tensor. 4 3 Weyl’s Unified Theory As powerful and profound as Einstein’s gravitational theory was, many felt it was only the beginning. To describe both known forces (the nuclear forces were not yet discovered), the EM field tensor had to be put in by hand. Many, including Einstein himself, sought a unified theory to explain both phenomena, preferably in a geometric fashion like GR. The first attempt to generalize GR to encompass EM, was proposed by Weyl three years later. Unhappy with Riemannian geometry, Weyl devel- oped his purely infinitesimal geometry which did not allow comparison at a distance. 3.1 Scale Invariance As is well known, in Euclidean geometry, translation of a vector preserves its length and direction. In Riemann’s geometry, the Christoffel connection guarantees length preservation, however, a vector’s orientation is path de- pendant. However, the angle between to vectors, following the same path, ispreserved under translation. Weyl wondered why the remnant of planar geometry, length preservation, persisted. After all, our measuring standards (rigid rods and clocks) are known only at one point in spacetime. To measure lengths at another point, we must bring our measuring tools along with us. According to Weyl, only the relative lengths of any two vectors (at the same point), and the angle between them, are preserved under parallel transport; the length of any single vector is arbitrary. To encode this mathematically, Weyl made the following substitution gµν(x)→λ(x)gµν(x). (9) Where the conformal factor λ(x) is an arbitrary, positive, smooth function of position. Weyl required that in addition to GR’s co-ordinate invariance, formulas must remain invariant under the substitution (9). Weyl called this agauge transformation . Remark 1. This was the first deliberate application of the gauge principle. In Riemann’s geometry, the metric is fixed up to a global scale factor. Weyl’s idea was to make that scale a local property of the metric. 5 Remark 2. The term gauge was introduced into mathematics and physics by Weyl during this period. Until now, its use in this paper has been purely from a modern perspective. In this setting, if a vector, vα, at a point P= (x1,x2,...xn) is parallel transported to the point P/prime= (x1+dx1,x2+dx2,...xn+dxn), then vα→vα+dvαwhere dvα=−vµ{α µν}dxν. (10) This transformation is identical to the Riemannian case, except the Christof- fel connection Γ has been replaced by a similar object, the conformal con- nection { }, which is also symmetric in the lower indices. 3.2 The Conformal Connection To find the conformal connection explicitly, consider two vectors, uαandvα, atP. Under parallel transport to P/prime, they become uα+duαandvα+dvα. By Weyl’s hypothesis, the relationship between the vector’s scalar products at each point is given by: (gαβ+dgαβ)(uα+duα)(vβ+dvβ) = (1 +dφ)(gαβuαvβ). (11) That is, the scalar products at PandP/primearenotequal, rather they are proportional. The factor of proportionality 1 + dφ, which is infinitesimally close to unity, distinguishes this geometry from Riemann’s. By expanding (11) up to linear differential terms, we have gαβuαvβdφ=dgαβuαvβ+gαβ(uαdvβ+duαvβ) (12) =dgαβuαvβ−gαβ(uα{β µν}vµdxν+vβ{α µν}uµdxν) =dgαβuαvβ− {αµν}uαvµdxν− {βµν}uµvβdxν, where we have used the substitution (10) for duαanddvβ, and then used the metric to lower the indices of { }. For the above relation to hold, for any vectorsuandv, we require that gαβdφ=dgαβ− {αβν}dxν− {βαν}dxν(13) =∂νgαβdxν− {αβν}dxν− {βαν}dxν, 6 where we have used the chain rule in the second line. Thus, dφis a linear differential form: dφ=φνdxν. Plugging this into the above, we find gαβφν=∂νgαβ− {αβν} − {βαν}. (14) By cyclicly permuting α,β andν, then subtracting the first arrangement from the sum of the other two, and finally applying the inverse metric gνλ, yields the conformal connection {λ αβ}= Γλ αβ+1 2(δλ αφβ+δλ βφα+gαβgνλφν), where Γλ αβis the usual Christoffel connection, derived from the metric. Thus, in Weyl’s geometry, the affine connection is doubly dependant; it is deter- mined by a) the metric tensor gαβand b) the covector φν. We will call this covector the length connection , as it relates the scales between two points on a manifold. Remark 3. Unless the φνare known, the conformal connection is not uniquely determined. Instead, there exists an equivalence class of connec- tions which preserve relative lengths and angles under parallel transport. 3.3 Electromagnetic Interpretation Now, if we perform the gauge transformation (9), since {αβν}=gαλ{λ βν}, then (14) becomes (suppressing the xdependance), λgαβφ/prime ν=λ∂νgαβ+gαβ∂νλ−λ{αβν} −λ{βαν} (15) =λgαβφν+gαβ∂νλ. From which we conclude, that the transformation gµν→λgµν⇔φν→φ/prime ν=φν+∂νλ λ. (16) Sinceλis an arbitrary smooth positive function, we can just as well set λ(x) =eα(x), for some function α(x). Now the gauge transformation reads gµν(x)→eα(x)gµν(x)⇔φν(x)→φν(x) +∂να(x). (17) In this highly suggestive form, Weyl’s gauge transformation of the length connection bears a striking resemblance to the gauge transformations of the 7 EM potential, Aµ. Weyl postulated that of all the possible conformal connec- tions in the equivalence class, only one had any relation to physics. Specifi- cally, the connection where φν=e γAν, whereeis the electron charge and γ is an undetermined constant. This is a most remarkable result. By allowing scales to vary between points on a manifold, Weyl could to formulate a unified theory of EM and gravity. The gravitational fields are encoded in the metric tensor, and EM fields are derived from the length connection form. What is perhaps even more remarkable is this theory sprung forth from a purely mathematical concept: Weyl’s infinitesimal geometry . GR, however, was derived from an entirely physical fact: the equivalence of inertial and gravitational mass. With this choice of the length connection φ, the Weyl connection is writ- ten (λ αβ) = Γλ αβ+e 2γ(δλ αAβ+δλ βAα+gαβgνλAν). It depends on both the gravitational fields, gαβ, and the EM potential Aµ. We can form all the objects from Riemannian geometry, such as the covariant derivative and curvature tensors, by replacing the Christoffel connection with Weyl’s. Rather than develop Weyl’s geometry further, let’s return to equation (11). Suppose a vector, vµatPhas the (squared) length l=gµνvµvν, then under parallel transport to P/prime, this length changes by dl=ldφ=le γAµdxµ. (18) If instead, this vector is (parallel) transported along the path Cto some distant point Q, then upon integrating (18), we find l=loee γ/integraltext CAµ(x)dxµ , (19) whereloislatP. So, a vector’s length is, in general, path dependant. At this point, a clever reader might raise the following objection. Suppose we have two identical clocks, at P. Letlbe the length of a time-like vector cor- responding to some unit of time. Now, transport the two clocks on different pathsC1andC2, which both end at Q. Letl1andl2denote the new values oflgiven by each clock, at the point Q. Then, by Stokes theorem, l1=l2if and only if /contintegraldisplay C1−C2Aµ(x)dxµ=/integraldisplay DFµν(x)dxµ∧dxν= 0 (20) 8 whereD=int(C1−C2) andFµν=∂µAν−∂νAµis the EM field tensor. Thus, in the presence of an EM field, the two clock rates will differ. As Einstein pointed out, the frequency of the spectral lines of atomic clocks would depend on the location, both past and present, of the atom. However, we know the atomic spectral lines to be quite definite, and independent of position. Einstein was able to refute Weyl’s theory, with this simple physical ar- gument. Weyl probably wished he had not sent his paper to Einstein to be published, since Einstein included this negation as a postscript. He, nonethe- less, admired Weyl as a brilliant mathematician, and was greatly impressed by his novel geometric ideas. Weyl, on the other hand, was not convinced and continued to develop his true infinitesimal geometry. He thought, It would be remarkable if in Nature there was realized instead an illogical quasi-infinitesimal geometry, with an electromagnetic field attached to it. Weyl’s gauge theory was paid little heed during the next decade. With the coming of the Quantum era, attention moved to the microscopic regime. 4 Gauge Theory Revived As pointed out in the introduction, Weyl’s unified theory, though flawed, was not utterly doomed. It was to find salvation during the development of Quantum Theory. 4.1 Quantum Mechanics In classical mechanics, a system, composed of a particle of mass min a potentialV, is said to be conservative if the total energy Eis a constant of motion. That is to say,d dtE= 0. Conservative systems are described by the Hamiltonian function H(x,p) =E=p2(t) 2m+V(t,x) (21) which is just the sum of kinetic and potential energies (bold variables are now 3-vectors). In Quantum Mechanics (QM), a particle is described by a complex-valued wave function ψ(t,x). For simplicity, we shall only consider non-relativistic QM, for spinless particles. Information is extracted from the 9 wave function, by acting on it with linear operators. Following Schr¨ odinger’s lead, let’s replace the momentum variable pby the momentum operator ¯h i/vector∇and the parameter Eby the energy operator i¯h∂t, where ¯his Planck’s constant, and /vector∇is the usual 3-dimensional gradient operator. Applying both sides to the wave function ψwe have found the Schr¨ odinger equation i¯h∂tψ(t,x) =−¯h 2m/vector∇2ψ(t,x) +V(t,x)ψ(t,x). (22) As a simple example, consider the plane wave ψ(t,x) =ei(k·x−ωt)(23) which has momentum p= ¯hkand energy E= ¯hω. It is not difficult to see that it solves the freeSchr¨ odinger equation, that is V≡0, andE=p2 2m. The probability of finding a particle in a given volume Ω is given by the formula Pψ(Ω) =/integraldisplay Ω|ψ(t,x)|2d3x. (24) |ψ|2has the natural definition then of a probability density . Wave functions are always normalized so that the integral of the probability density over all space equals unity. However, this does not determine ψuniquely. For example,ψ/prime=eiαψ, for some real number α, (look familiar??) has the same probability density as ψ. Thus, the states ψandψ/primeare equivalent. 4.2 Phase Invariance Because of this equivalence, we say that wave functions in QM have a global phase symmetry . Recalling the plane wave model, we notice that multiplying it byeiαis equivalent to adding a phase αto the argument. However, we can go one step further. If we believe the gauge principle (and why not?), then we can make this phase invariance a local property of the matter wave by sending α→α(t,x). After all, we still have |ψ/prime|2= (e−iα(t,x)ψ∗)(eiα(t,x)ψ) =|ψ|2. But this leads to complications. Consider a function ψwhich solves the free Schr¨ odinger equation, and perform the local phase transformation ψ(t,x)→ψ/prime(t,x) =eiα(t,x)ψ(t,x). (25) 10 The Schr¨ odinger equation should remain unchanged, since both states are equivalent. However, the transformed equation becomes (suppressing ( t,x) dependance) i¯h∂t(eiαψ) =−¯h2 2m/vector∇2(eiαψ). (26) Clearly, when the derivatives act on the exponentials, new terms will be introduced. If we consider any single spacetime derivative, then we have ∂µ(eiαψ) =i∂µαeiαψ+eiα∂µψ (27) /negationslash=eiα∂µψ. The second line is what we want. We have encountered a similar problem before, regarding co-ordinate invariance for GR. Then, we had to modify the partial derivative, making it covariant. Instead of changing co-ordinate frames, this time we are changing the phase. Let’s attempt a similar manip- ulation. So, we seek a covariant derivative Dµ, such that under the transfor- mation (25) Dµψ→eiαDµψ, (28) soDµmust transform as, Dµ→D/prime µ=eiαDµe−iα. (29) Consider, then, the object Dµ(x) =∂µ+iAµ(x) (30) whereAµ(x) is some covector field. Then under the local phase rotation, (30) tell us that this covariant derivative transforms as Dµ→D/prime µ=∂µ+iA/prime µ (31) (since the partial derivative operator is position independent). So it is this covector field Aµthat must vary under phase changes. To find an explicit form for this variation, we simply require that (28) and (31) coincide. That is to say, eiα(∂µ+iAµ)ψ= (∂µ+iA/prime µ)eiαψ (32) =i∂µαeiαψ+eiα∂µψ+iA/prime µeiαψ. 11 Which implies A/prime µ=Aµ−∂µα. (33) Thus, the modified Schr¨ odinger equation i¯h(∂t+iA0)ψ(t,x) =−¯h2 2m(/vector∇+iA)2ψ(t,x) (34) is invariant under the simultaneous local transformations ψ(t,x)→eiα(t,x)ψ(t,x)and Aµ(t,x)→Aµ(t,x)−∂µα(t,x). (35) 4.3 Relation to Electromagnetism The results of this local phase invariant are quite profound. We were forced to change our momentum operator into¯h i(/vector∇+iA). Thus, changes in a par- ticle’s phase, which alters the 1-form Aµ, result in changes in its momentum. According to Newton’s second law, F=m¨x=˙p (36) ( ˙ signifies time derivatives) there must exist a force which performs these changes. That force is none other than EM. The equation i¯h∂tψ(t,x) =−¯h2 2m(/vector∇ −iq ¯hA)2ψ(t,x) +qφ(t,x)ψ(t,x) (37) whereqis a particle’s charge, is empirically known to govern the motion of a charged particle in an arbitrary EM field. This is equivalent to equation (35) if we multiply αandAµbyq ¯h. The gauge transformations now read ψ(t,x)→eiq ¯hα(t,x)ψ(t,x)and Aµ(t,x)→Aµ(t,x) +q ¯h∂µα(t,x).(38) So, local phase invariance introduces EM interactions. Interestingly, the above transformations are identical to Weyl’s earlier gauge transformations after a) replacing the metric gµνby the wave function ψ, and b) setting the undetermined constant γto¯h i. The first substitution tells us that EM is a phenomenon that accompanies matter fields, and not, as Weyl thought, the spacetime metric. Changing the constant γfrom real 12 to imaginary, takes Weyl’s conformal factor from the positive real axis to the unit circle in the complex plane. Weyl’s non-physical path dependant vector lengths become the well proven path dependant matter-wave phases. Historically, these correlations were first pointed out by Sch¨ odinger and London in the 1920’s, though only in a rather tentative manner. Then, in 1929, Weyl published the paper Electron and Gravitation which introduced many now fundamental concepts, including a derivation of EM from the gauge principle. Before its release, Weyl published a short summary to which Pauli, upset by the mathematician’s intrusion into physics, replied, I admire your courage; since the conclusion is inevitable that you wish to be judged, not for your success in pure mathematics, but for your true but unhappy love for physics. However, after reading the whole article, Pauli wrote back saying, Here I must admit your ability in Physics. Your earlier theory withg/prime ik=λgikwas pure mathematics and unphysical. Einstein was justified in criticizing and scolding. Now the hour of your revenge has arrived. 5 Gravity as a Gauge Theory The procedure given above, to construct locally gauge invariant systems, may be generalized to more complex symmetries. Doing so results in the introduction of more complicated fields, and hence, new forces. We have briefly seen how GR is a gauge (co-ordinate) independent theory. Let us develop this more formally. Note: To properly understand the gauge structure of GR requires much more mathematical machinery than this paper shall develop. Instead, this section shall sketch the proper approach to formulate a gauge invariant theory of gravitation. 5.1 Vierbein Formalism The Minkowski spacetime metric, as a matrix, is ηmn=diag(−1,1,1,1). Latin indices will be used to denote co-ordinates in this Minkowski basis, while Greek indices are reserved for arbitrary reference frames. According 13 to the Equivalence Principle, we may always choose our reference frame, at each point in spacetime, as a Minkowskian one, so that there appears to be no gravitational force! By re-expressing the Minkowski basis {xm}in terms of a general one {xµ}, we see the effect of gravity. By formulating our physics in the {xm}basis, they are independent of any relabelling {xµ}. This is the gauge invariance of gravity. (Proceeding in this manner is particularly useful for constructing spinors in curvilinear co-ordinates.) The effects of gravity are contained in the changes of {xm}from point to point. Expressing this basis in terms of the general one, we have dxm=hm µ(x)dxµwhere hm µ(x)≡∂µxm. The transformation matrix hm µ(x) is called the vierbein . Likewise, we define the inverse vierbein by dxµ=hµ m(x)dxmwhere hµ m(x)≡∂mxµ. We see, then, that an arbitrary spacetime metric can always be written gµν(x) =ηmnhm µ(x)hn ν(x). (39) Lorentz transformations Λn m(rotations in spacetime) are equivalent to a change of basis. By requiring invariance under local Lorentz transformations, we must introduce the covariant derivative for the {xm}basis, which depends on some new field ωµ, Dm=hµ mDµ=hµ m(∂µ+iωµ), (40) withxdependance suppressed. To understand this operator in curved space- time, consider the expression hm µDνvm. Though the calculations are beyond the scope of this paper, the end result is that hm µDνvm=∂νvµ+ Γλ µνvλwith Γλ µν≡hm µDνhλ m. (41) From which we may define the familiar ∇νvµ≡∂νvµ+ Γλ µνvλ. If there is no torsion (Γλ µν= Γλ νµ), then Γλ µνis the Christoffel connection. One important consequence of this approach is to free the covariant derivative 14 from the notion of parallel displacement. Instead the more fundamental gauge principle is used. Again, this section is by no means a rigorous derivation of GR from the gauge principle, rather, it has been presented to highlight the analogies between EM and GR as gauge theories. The inclined reader should consult [3] for a complete account. Also suggested is Utiyama’s paper, Invariant Theoretical Interpretation of Interaction , translated in [2], where a general method for constructing gauge invariant interactions is developed, with EM, Yang-Mills and GR as worked examples. Though published after Yang and Mills historic paper, it was in fact, written one year prior. 5.2 Comparison of Gauge Groups Gauge theory has a very natural formulation in terms of groups. Recall that a group,Gis a set of elements xsatisfying i)∃e/epsilon1Gsuch thatex=x,∀x/epsilon1G ii)∀x/epsilon1G,∃y/epsilon1Gsuch thatyx=e iii) Ifx,y/epsilon1G , thenxy/epsilon1G Some known aspects of the gauge group structures of EM and gravitation will be presented to highlight the differences between the two as gauge theories. Recall the gauge principle states that ”systems invariant under a global group of transformations, should remain invariant when that group is con- sidered locally”. For EM, that group is, of course, the phase transformation eiα(t,x). Each element may be represented as a point on the unit circle in the complex plane, formally this group is known as U(1). It has a commutative structure and a compact topology. Also, it depends on a single parameter α(t,x). On the other hand, GR is invariant under change of co-ordinates; that is rotations and translations in spacetime. These transformations form a group, as well, called the Poincar ´e group (PG). There are 10 indepen- dent parameters: six for the Lorentz transformations and four for spacetime translations. Although translations are commutative, the (generalized) rota- tions are not, thus PG has a non-commutative structure. Furthermore, PG is non-compact since translations are unbounded. Some other notable distinctions between the forces are the following. In EM, the nature of interactions is determined by the sign of the charge q, while for gravity all matter is attractive. Apart from the charge, all EM 15 objects arise from the 1-form (also called the connection) Aµ, while in GR it is not the connection Γλ µνbut the 2-form gµνwhich plays the primary role. The most remarkable fact, though, is the similarity between the two forces. The gauge principle is truly fundamental to the nature of each force. 6 Conclusions Perhaps the most important result is that we have found the necessary con- dition to make symmetries local. It appeared in GR, and again for EM: the covariant derivative. The gauge principle, presented near the outset of the paper, should include the provision: that partial derivatives be replaced by covariant ones, which de- pend on some new vector field. These fields correspond to the four known force fields. By generalizing the procedure of section 4.2, we may develop gauge invariant theories for the nuclear forces as well. Clearly, the gauge principle’s role in modern physics is key. From it, we can determine the nature of the force fields, as well as their interactions with matter. It serves as the base upon which unification theories are built. Whether a unified theory for all interactions will ever be developed is hard to say. That the gauge principal will play a central role in it is hard to deny. 7 References 1. C. Cohen-Tannooudji, et.al, Quantum Mechanics . John Wiley and Sons, Toronto, 1977. 2. L. O’Raifeartaigh, The Dawning of Gauge Theory . Princeton University Press, New Jersey, 1997. 3. P. Rammond, Field Theory: A Modern Primer . Addison-Wesley Publish- ing, Don Mills, 1989. 4. E. Scholz, Hermann Weyl’s Raum-Zeit-Materie and a General Introduc- tion to His Work . Birkh¨ auser Verlag, Boston, 2001. 5. H. Weyl, Space Time Matter . Dover Publications, New York, 1950. 16