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symmetry has become the foundation of physics
Symmetry in physics is coded in terms of the Group concept as a Group of symmetry operations. The central feature of this group is the reversibility of operation. There are three fundamentally important physical symmetries of this type and their existance is justified by their presence in the three theoretical structures of Newton, Maxwell, Einstein and in their ramifications. These three symmetries are, Charge Conjugation (C), Parity Inversion (P) and Time Reversal (T). In C-symmetry a physical system as represented by these theories remains unchanged when the signs of all the charges are reversed, in P-symmetry the system is insensitive to a reversal of all left and right handed components and in T-symmetry the system will still work with no change if it is made to run backwards in time by reversing the sign of the time parameter.
In the summer of 1956 C. N. Yang and T. D. Lee at Brookhaven National Laboratory on Long Island showed that the P-symmetry need not hold in weak interactions such as the beta-disintegration of a neutron into a proton, electron and anti-neutrino. In other words the physical world at the subatomic level is weakly homochiral (or uni-rotational). In beta-disintegration the electron always comes out in the same direction with respect to the spin sense of the nucleus. If the nucleus is spinning anticlockwise with respect to the observer then the electron is always ejected away from the observer. Feynman in his famous book 'The Character of Physical Law', had this to say about this weak asymmetry between physical left and right.
'.....the remarkable feature of the fact that we can distinguish right and left is that we can only do so with a very weak effect, with this beta-disintegration. What this means is that nature is 99.99 per cent indistinguishable right from left, but there is just that one little piece, one little characteristic phenomenon, which is completely different, in the sense that it is completely lop sided.This is a mystery that no one has the slightest idea about yet'.
Both experiment and theory later indicated that C-symmetry is also violated during weak interactions. This violation is such that during the beta-disintegration of an antineutron, positron is ejected in a direction opposite to that of electron ejected during the the beta-disintegration of a neutron, if spin senses of both nucleons are the same. So, both C and P symmetries are individually violated by weak interactions in such a way that CP-symmetry is held. However in 1964 evidence emerged that CP violations also occur in weak interactions. It is accepted now that in general only CPT symmetry holds in weak interactions. But C, P and T symmetries continue to individually hold in the theoretical representations of gravitational, electromagnetic and strong interactions.
The significance attributed to individual C, P and T symmetries is due to their presence in the three theories mentioned above, particularly Maxwell's Theory and Relativity. These theories satisfy the Lorentz Symmetry Group of operations as a fundamental necessity and therefore the conditions under which the symmetries hold are also simply those which are characteristic of the Lorentz Symmetry Group. This group focuses only on the magnitude of the motion of matter. The direction of motion has been rendered arbitrary by the Group. This rendering causes a strategic reduction of the pair of observers in the original form of special relativity to just one observer. However such a reduction is unreal as in the true spirit of Special Relativity the observers must remain inseparably and irreducibly locked to each other. For, the well known experimentally verifiable special relativity predictions which inspire scientist and layman alike involve both the observers. This pairing of observers in special relativity is a simplified physical realisation of an abstract concept of pairing of vectors which is the basis of the Balance Foundation of Physics given in the Theory Pages. That pairing provides the infrastructure for proper direction which, unlike conventional direction, is independent of coordinate systems. So, whilst a single vector can carry with it only the length-information, for the direction-information it has to pair form with another vector. A fundamental consequence of this pairing is the non-interchangeability of the two vectors in 4-dimensional space-time. It is precisely this non-interchangeability which manifests as the violations of individual C, P and T symmetries. When the vectors are interchangeable the pairing is lost and the vectors effectively reduce to a single vector.
As a rule the two vectors are interchangeable only in 2-dimensional space-time where the two vector fields generated by the vectors always produce meshing congruences. In higher dimensions where such meshing need not take place the vectors can remain non-interchangeable. It will become clear in the Theory pages that the physical conditions necessary for meshing to take place in 4-dimensional space-times is that only gravitation and Maxwellean electromagnetism are present as material interactions. So, at a time when the only known interactions were these two it is not surprising that the reduction of the pair of observers in special relativity to just one observer in accord with Lorentz symmetry group remained firmly in place. Somehow even to this day this lone observer has continued to carry the burden of transition from special to general relativity. In this way while special relativity actually denies a physical significance to a single observer, in a generalized setting this denial is withdrawn. The argument here is that the global features of a metric tensor field alone are sufficient to justify the lone observer created by the Lorentz symmetry group. However, this argument begs the question of establishing a sound theoretical procedure that leads to the formulation of the present field equation for the metric tensor. For at present there is none, at least none consistent with the integrity and the beauty of the basic framework of relativity as admitted by Einstein himself. According to Balance it is precisely the instigation of the lone observer by the Lorentz symmetry group that stands in the way of this formulation.
A crucial consequence of the use of Lorentz symmetry group is the presence of rest frames. But where and how does one find a rest frame in a truely fundamental setting? Even at the temperature of absolute zero when classical motion stops, quantum mechanical motion, which is likely to be closer to the real thing, persists. According to Balance the simplest and the most fundamental form of real motion is what the pair of observers in special relativity represents. It then follows that, somewhat paradoxically, rest and motion are only meaningful in relation to each other and so, here we find a beautiful coexistence between rest and motion and it is this coexistence that Lorentz symmetry group disturbs. This coexistence represents a state of meaningful dynamic interaction between motion and its surrounding. Put simply a particle and its surrounding form an unbreakable whole which has a perfectly defined balanced structure. Accordingly when a particle falls freely in spacetime it is not just slipping through a groove that it cuts in spacetime but is engaged in a balanced dynamic negotiation with its surrounding.
Because of the incompleteness of the present physical theories, as outlined above, there are fundamental discrepancies between the theoretical world of physics and the real world in which we live. The real world, not just the world of weak interactions, in defiance to what Feynman said above, mercilessly violates C, P ant T symmetries and at present these violations are naively considered as due to accidents of nature. Firstly the real world that we know of is almost completely made up of matter. Secondly nothing in the real world displays left-right symmetry; according to Pauli exclusion principle even two 'identical' electrons are not interchangeable which, in a manner of speaking, means that a real electron distinguishes between left and right. Finally the world inexorably marches forward in spacetime. To consider a few real life examples of the violation of P-symmetry, which is the more questionable of the three violations in everyday life, right handed people far out number the left handed. Also there are many fundamental features of the biological world which distinguish between the left and the right or exhibit homochirality. All twenty biological amino acids or proteins, with the exception of a simpleton which does not exhibit chirality, are levorotatory. All sugars are dextrorotatory and so also are the nucleic acids, RNA and DNA. Critics may argue that if amino acids or sugars are made in the laboratory both dextro and levo types are formed in equal numbers and hence it is likely that the real world is P-symmetric to a very high degree at the very least. But left-right non-interchangeability in the physical world occurs at a 'field' level, or nuclear level, which is higher than that occupied by matter during chemical reactions and it is this chemical state of matter which displays C, P and T symmetries. So, it appears that life is not just chemical, it is, at the very least, nucleic as well to a degree even above and beyond what is implied by the nucleic acids of DNA, RNA and their complements.
The view that it is symmetry breaking which produces an asymmetry is actually a bitter pill to swallow as then asymmetry has to be painfully constructed out of an available symmetry and an unavailable antisymmetry that has to be dragged in from somewhere. What is natural and almost obvious is the view that it is an asymmetry which is there in the first place and it is the conditional breaking, or splitting, of this asymmetry which produces symmetry and antisymmetry. As the Theory pages will show it is precisely this view that is in accord with Balance.
There is a set of 'concrete' symmetries which are truly fundamental, unlike the C, P and T symmetries. The symmetry object in this case is the body of the laws of physics itself. The symmetry arises from the fact that the laws of physics are independent of references for temporal, spatial and orientational positions. In other words the 'spirit' of physics functions independently of origins in time and space, and also independently of a reference for orientation. Putting it more simply the laws of physics are the same today as they were yesterday, the same elsewhere as they are here, and the same whether the physical world flows in that direction or this. Putting it precisely laws of physics are independant of differentiable transformations of systems of coordinates.
Let's look at this wonderful symmetry from another perspective. All that in the previous paragraph may be summed up by saying that the laws of physics are based on a pair formation between any two nearby events in spacetime. As pair formation is non-interchangeable the physical world flows from one event to the other irreversibly. The conventional velocity vector results from a limiting process that draws the two events together. But the resulting loss of pair formation has to be compensated somehow and the result is the presence of a pair of velocity vectors in place of just the one vector which is used in physics at present. That pair of velocity vectors which is the forerunner of linear motion and spin is precisely the basis of the Balance Foundation of Physics.
The principle features of Symmetry Group (of operations), in their simplest form, are contained in circular symmetry. A plane circle remains unchanged if we rotate it through an arbitrary angle θ about a perpendicular axis passing through the centre. it also remains unchanged if we reverse the rotation completely. Thus every positive angle of rotation can be paired off with its negative and vice versa. Consequently, the rotation operations which generate circular symmetry are antisymmetric. Next a rotation through an angle θ1 followed by another through an angle θ2 is equivalent to a single rotation θ1 + θ2 . This feature, which is generally referred to as 'closure of a group of operations', is quite obvious for these two-dimensional rotations, but it is far from obvious for rotations in higher dimensions and for operations of a general nature. 'Closure' sets Symmetry free of operational sequences, or paths between two positional configurations and it also causes Symmetry to close in on itself and become insular. The two axiomatic features of 'reversibility' and 'closure' so simply illustrated by circular symmetry, along with two other axiomatic features which will be given below, form Symmetry Group of operations. The following is a formal description of Symmetry Group features.
1. The characteristic feature of Symmetry Group is the reversibility of an operation. As reversibility is characteristically antisymmetric, Symmetry Group contains both symmetry and antisymmetry features. However owing to the arbitrary character of reversible operations, antisymmetry remains arbitrary and therefore also independent of symmetry. Here antisymmetry is used in a general sense and refers to the pairings, operation and its reverse or inverse operation.
2. Symmetry Group engineers a self existing, or self mapping, feature of an object with respect to a set of operations (on the object) which is independent of any particular configuration of the object and also any particular path taken by the operations between one configuration and another. The symmetry feature of the object is then free of arbitrary references and influences to the extent covered by the symmetry group operations and thus possesses a rational objective identity.
Continuing with the exploration of symmetry features in relation to circle, now consider a radial line. Along this line θ (or direction) remains constant provided we use the convention that r lies within 0 and ±infinity. The admission of both positive and negative values for r prevents θ from making a step change at the origin. It is clear that these values of r measured along the radial line also satisfy symmetry group axioms. However, although the radial line possesses symmetry it is the circle that appears to be truly symmetric and not the radial line. This aesthetic preference which ignores formal mathematical rigor of symmetry group can be explained as follows.
1 Circular symmetry results from a local operation which is centred around a point and it so happens that 'linear' symmetry of the radial line results from a global operation similar to the target seeking of a guided missile. Now in accord with the principles of mathematics and physics, the rules of operation of the physical world in their fundamental form, or the fundamental laws of physics such as the conservation principles, are local and not global. In a manner of speaking the 'fate' of the global is sealed in the local somewhat like the impressive features of a giant redwood tree are 'preprogrammed' into its tiny seed. However, we shall later find that local and global are not that surgically cleanly separable as 'quantum non-locality' indicates; it is just that local is our starting point of rational (or analytical) enquiry and global, its progression. A simple analogy would be zero and infinity where one is the inverse of the other and only the zero is the rational of the two. It is curious that zero is also a little circle.
2 The invariant in line symmetry is angle θ and therefore another line is involved in addition to the line we are focusing on. So, in a manner of speaking line symmetry is not free of external references. Now it may be argued that a radial line considered as a geodesic carries its own reference direction with it and therefore line symmetry in this respect is free of external influences. However geodesic is a concept that is contingent on high level concepts such as the 'covariant derivative' which in turn depend on the basic axioms of symmetry. Putting it simply, a geodesic is what becomes of line symmetry when it puts on a ceremonial robe.
The formal structure of symmetry group consists of a set of symmetry operators, a, b, c etc. and a rule 'o' which combines any two of the operators to produce an operator which also belongs to the set. Addition and multiplication are the usual examples of 'o'. The operators obey the associative law
a o (b o c) = (a o b) o c
There is a unique operator s belonging to the set which combines with any operator, a, (including itself) thus
s o a = a o s = a
Each and every operator a has an inverse a' such that
a' o a = a o a' = s
One of the most prominent symmetry groups is the Lorentz Group in 4-dimensional spacetime. Here the operators are orthogonal matrices of numbers representing rotations in spacetime of a four-dimensional velocity vector and the orthogonality of the matrices ensures that the magnitude of the vector remains invariant. The rule is matrix multiplication and the unique entity is the identity matrix all of whose elements are zero except those along the leading diagonal which are of unit value.
The most general symmetry group is the general linear group of all non-singular n x n matrices dented by GL(n,R) where n is a positive non-zero integer and R denotes that the matrix elements are real. Symmetry objects in this case are tensors, or the rudimentary building blocks of the geometrical representation of the physical world.
Newton's laws of motion can be interpreted in terms of symmetry and antisymmetry. Consider his third law which describes the state of interaction between two bodies as an action and a reaction that are equal and opposite. Accordingly there is a plane of symmetry between the forces of action and reaction which form a line of antisymmetry at right angles to the plane. Next consider his first and second laws which describe the state of existence of a single body in terms of a momentum field. This field, in its complete form, can only be accessed from the framework of relativistic physics which is more fundamental than that of Newton's physics whence it is referred to as a 4-dimensional 1-form field. This 1-form field (say, u), generates an anti symmetric 2-form field (say, f) as follows.
In this expression {x0, x1, x2, x3} are time and space coordinates respectively. For a given u, this f corresponds to the pure spinning component of u in spacetime. If this motion is set to zero, then u would represent only a pure translatory motion in spacetime and we recover the first and second laws, as discussed below. Spinning motion then becomes an artifact requiring the specification of an arbitrary axis of reference. This artifact is angular momentum.
Now for i = 0, j = 1,2,3 we get
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which is the same as
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It is this equation that takes the form
Force = Rate of Change of Momentum
which is Newton's second law. The vanishing of the three remaining components of f is the condition which reduces motion to a pure rectilinear one.
In conclusion, Newton's physics is built upon a fixed configuration of symmetry and antisymmetry of interaction between two bodies and negation of an antisymmetry in the case of the motion of a single body.
The progressive diversity of life thrives on action and reaction not being equal and opposite. If a good deed from a person A to a person B is simply returned by B to A the scope of good becomes limited. A bad deed, or an assault, or an insult from one person to another is defused if the receiver does not respond with equal measure but chooses to ignore the assailant.
Loosely speaking physical symmetry means preservation of a form or a property, against a related change or operation and according to Noether's theorem there is a one-to-one correspondance between any differentiable symmetry operation in physics and a conserved physical parameter (usually referred to as a conservation principle). The abstract symmetry group GL(n,R) which preserves tensor character is a group of differentiable symmetry operations outside of which no physical theory based on geometry can maintain its integrity. Reduced to its nuts and bolts the presence of this group implies that temporal and spatial positions and the flow direction of the world are free of absolute references. According to Noether's theorem this three-fold differentiable freedom exists in association with a three-fold conservation principles of energy, linear momentum and angular momentum. This group of principles is present only as three separate principles in Newton's physics. Although only the conservation of energy is unconditional in Newton's physics, since forces always occur as pairs whose members are equal and opposite, a system can always be found, at least in theory, which satisfy the conservation principles of momenta.
In post-relativistic physics Newton's 3-dimensional space and 1-dimensional time became a 4-dimensional spacetime characterised by a symmetric metric tensor field. Consequently 3-velocity became a 4-velocity which together with the metric tensor produces a 4-dimensional 1-form, in which Newtonian forms of energy and linear momentum become integrated into a 4-dimensional form. There is a scalar associated with this 1-form (the inner product between 1-form and 4-velocity) and it represents particle rest mass. Therefore Newtonian conservations of energy and linear momentum are really fragmented forms of a more fundamental conservation of energy that manifests as rest mass the conservation of which is vital to the stability of matter. So much for the relativistic transformation of Newtonian energy and linear momentum, but what of the transformation of angular momentum which owing to its quantum character stands in stark contrast to linear momentum which has a continuum character? Presently it appears as a 2-form and together with the linear energy-momentum 1-form it represents what is referred to as the Poincare Group of motions. This group is the Minkowskian equivalent of the group of motions of a rigid body in 4-dimensional Euclidean manifold. Different fundamental particles correspond to different representations of the Poincare Group which result when corresponding masses and charges are put in place. Poincare Group is 10-dimensional and it so happens that in Balance, motion is represented by a pair of 4-vectors together with a pair of scalar parameters which therefore possesses exactly ten variables. Thus in Balance the conventional 'ball and chain' representation of motion which keeps a particle tied to 'terra firma' becomes replaced by a pair of 'wings' that allows it to fly freely in an 'ethereal wind'. What this ethereal wind exactly is given in precise mathematical detail in the Theory pages.
Since according to the first law of thermodynamics energy can neither be created nor destroyed, there is also an absolute global form of conservation of energy which is process- or system-independent and this global form raises two inevitable questions. How much of energy is there in the physical world and who or what put it there in the first place? It has been suggested that the global energy is zero and therefore the second question is superfluous. This zero energy consists of a negative gravitational energy and an equal amount of positive energy in conformity with Newton's vision that everybody (or every energy because of mass-energy equivalence) attracts every other body (or every other energy). Viewed in this manner the concept of global zero energy looks mysterious and ill-founded as, at present, there are dark and mysterious energy forms operating in the background. Also in a rational view of the physical universe global features are best left to themselves to develop from local features as is the practice in mathematics.
According to Balance Nature designs the tapestry of the physical world in such a way that each and every 4-dimensional event in the physical universe carries equal amounts of positive and negative energies which may be referred to as inertial and kinetic respectively. The mathematical details of how she carries out this design is given in the Theory Pages. So, at each and every event there is a perfect balance between two fundamental forms of energies and there are two inspiring ways in which this condition of balance can be expressed. First of these is to say that at each and every event there is a state of free fall, which is a complete generalisation of the gravitational free fall that affected Einstein so deeply. The second is to say that when all things are considered each and every event in the physical universe is complete by itself.
As every event carries zero energy the total energy in the universe is indeed zero. But that result, this time, is arrived at on the basis of local considerations only.
Energy is the only conservation principle that matters in Balance. Although it is zero every where-when, it is also the inner product of several tensors ( a tensor is a mathematical representation of a basic geometrical element) and hence the supporter of a very rich structure. The complete mathematical details of this structure along with detailed mathematical descriptions of the various assertions made above are given in the Theory Pages.
the former represents symmetry and the latter, antisymmetry
The divergence of a vector field, being a scalar, is symmetric with respect to a reversal of the handedness of a coordinate system and the curl, being a polar vector, is antisymmetric with respect to this reversal.
In the framework of relativity Maxwell's electromagnetic theory reduces to two sets of equations. The first is automatically satisfied if the electromagnetic field originates as a 1-form field. The second set relates this 1-form field to a charge-current density vector field. Given this vector field Maxwell's theory determines the 1-form field, or so it seems. However, if the history of the evolution of Maxwell's theory is considered, an antipodal picture emerges. In this picture the 1-form field is the primary factor, and it is used to define the charge-current density vector field, according to the macroscopic experimental work on static electric fields carried out by the French engineer Charles Augustin de Coulomb (1736-1806) and the experimental and theoretical work on electromagnetic fields carried out by the French physicist and mathematician André-Marie Ampère (1775 –1836). Accordingly, first set of Maxwell's equations amounts to a declaration of the existence of a 1-form field as part of the fundamental theoretical structure of the physical world, and the second set of Maxwell's equations reduces to an identification of the divergence which is derived from this 1-form field with a charge-current density vector. From Balance point of view this identification amount to a definition of the charge-current density vector, or the divergence of the electromagnetic field, in terms of the 1-form field. But it is also a physically realistic definition, macroscopically, as charge is conserved. The curl which corresponds to the divergence vanishes automatically as a characteristic feature of 1-form fields.
an event and a set of coordinates
At present an event and its associated set of coordinates are seen as wholly independant of each other. The underlying principle is 'General Covariance' and it is a statement of the fact that physical laws can be formerly written without reference to any specific system of coordinates. Thus General Covariance underpins a limiting case of pair formation between two entities (of the same type) in which each exist in a state of freedom from the other. But Balance decrees that the opposite extreme in which the two entities are in complete coincidence should also be equally represented. In that case the tensor elements in terms of which physical laws are expressed has a preferred coordinate system. Therefore, in Balance, the principle of general covariance is more meaningfully interpreted thus: ‘a class of physical phenomena consonant with a particular coordinate system can be translated into any other coordinate system provided a differentiable map exists between the two sets of coordinates’. This interpretation opens the door for coordinate systems to assume a deep catalytic role in physics. In this role a coordinate system truly becomes what makes it possible for an act of observation or analysis to be initiated or to take place speedily without itself disturbing that act. But it also emulates its counterpart in chemical reactions and therefore for a given class of phenomena not just any coordinate system will do to start with.
This view of pair formation between an event and its coordinates also puts the schools of E. Cartan and Princeton on an equal footing so that they are now no longer alternatives but are two different phases of one process. According to the first school, during a 'coordinate' transformation it is the event that changes while the coordinates remain fixed, and according to the second school it is reverse that takes place. (J L Synge, Relativity: The Special Theory, John Wiley & Sons, Inc, New York, 1964, Ch. IV, p 75)