A Heuristic, Non-Mathematical Approach to Superstrings/M-Theory

By David Sears Schroeder

Throughout most of string theory's evolution the concept of supersymmetry, and its derivative supergravity, have provided an indispensible underpinning. Without supersymmetry M-Theory: the latest incarnation of string theory, would completely lose its logical cohesion. The original point particle based standard model also benefited greatly from the inclusion of supersymmetry. And, supersymmetry figures into an alternative approach to string theory, in which supersymmetric partners are postulated to constitute a substructure to the known particles of the standard model. Finally, supersymmetry is the only principle that allows theorists to quantize the gravitational field, and thus unify gravity with the other three forces.

Clearly, supersymmetry is a vital ingredient of the most promising theoretical programs to advance the standard model beyond its present plateau, irrespective of which approach ultimately proves correct. But its application to theory, apart from the half integer spin displacement for bosons and fermions, has become mired in a labyrinth of esoteric mathematics, stifling progress. This circumstance should prompt us to ask whether there might be another way to look at the supersymmetric operation that is not so dependent on a purely mathematical formulation. A different perspective on a problem often leads to fresh insights, facilitating the resumption of theoretical advance.

A major clue to unraveling the physical meaning of supersymmetry may be present within the structure of the standard model itself. In the early decades of the 20th century the renowned British physicist Paul Dirac made two important predictions that deal with fundamental symmetry translations: 1) That every particle has a counterpart with opposite charge and/or spin, and 2) That quantized units of magnetic charge might exist, known as magnetic monopoles. The first prediction has been abundantly demonstrated, while the second has yet to be verified.

Despite the absense of detection, the case for the existence of monopoles is greatly strengthened by the fact that they constitute an essential feature of a wide class of Grand Unified Theories (GUTS). Furthermore, Dirac found that the existence of a single monopole automatically imposes quantization on all electric charges. An upper limit is placed on monopole abundance by the Parker bound, which relates to the observed strength of our galaxy's magnetic field. This, combined with their enormous estimated mass ~ 1015 GeV, may make monopoles rare quarries, possibly accounting for the lack of detection with the limited searches conducted so far.

The existence of antimatter, and the assumed existence of magnetic monopoles, implies that the fundamental fields of nature, and the particles associated with them, are inherently mutable. Additionally, the symmetry translation associated with magnetic monopoles is clearly more complex than that associated with antimatter. With antimatter the symmetry translation involves only the polarity inversion of the electric field (and/or spin reversal), for a given type of particle. However, with magnetic monopoles the very roles of the electric and magnetic components of the unified electromagnetic field become transposed, while the basic field relationship between these components remains intact. That is, the motion of a magnetic charge gives rise to an electric dipole field in exactly the manner that moving electric charges give rise to a dipole magnetic field. In short, the monopole acquires the role of an electric charge, but couples to the electromagnetic field via its magnetic charge.

This pattern of increasingly complex symmetry translations within the electromagnetic field begs the question whether this trend might continue to even higher order symmetry transformations. In particular, since experimental and theoretical evidence firmly points towards the existence of a superforce unifying all four fundamental forces, plausibly these components of the superforce might, at some energy level, also be able to transpose roles with one another. By this is meant a symmetry transformation in which the fundamental structure of these fields remain intact, but each field now couples to the other's type of force - a direct linear extrapolation of the symmetry between electric charges and magnetic monopoles. It's proposed that a transformation of this type constitutes the essence of a supersymmetry type operation. In the case of two fields having widely different coupling strengths, a reciprocal force trade would be required to maintain symmetry.

If we regard the role exchange of any two of the four forces at a time as one permutation, this would allow for twenty four permutations of field arrangements: (N!, where N = 4). Modifying this appreciation, however, is the fact that the electromagnetic and weak forces manifest their union in the electroweak synthesis, via the W's and Z0, at the very low energy of 90 MeV - below the supersymmetry scale, - suggesting these two fields should be treated as one. In that case, N reduces to 3, and the total number of field permutations drops to 6, which by a curious coincidence matches the number of field theories embraced within M-Theory. One of these six field arrangements would presumably correspond to the set of particles and fields in our world, while the remaining five would represent universes with their own unique sets of particles and forces.

This numerical coincidence may or may not be significant, but its tantalizing to suppose that the six field arrangements derived throuth the permutation process correspond to the five 10 dimensional string theories, plus 11 dimensional supergravity, that together form the substructure of M-Theory. If so, this field permutation approach would constitute a simpler way of viewing this complex, higher dimensional theory arrived at through prodigious mathematical labors. The field permutation mechanism itself could then be seen as duplicating the action that the duality mechanism performs in linking the six subtheories of M-Theory to one another. But, the fly in the ointment here is that string theorists are assuming that duality implies the different subtheories within M-Theory are describing an identical underlying theory.

Oddly enough the field permutation process leads to the conclusion (at least in the case that will be examined shortly) that universes with intrinsically different field structures have identical properties when perspective is taken into account. The idea would be that a person living in a universe with a different field structure from ours would perceive all the laws of physics to be identical to our own, provided he himself consists of the same unique matter comprising that universe. From his perspective, the particles that make up our universe would be quite massive, and he would conclude that a symmetry breaking was responsible for the great mass of these particles.

There is a curious parallel to this idea in special relativity. The laws of physics are required to be identical in every reference frame. Yet, observers in reference frames moving with respect to one another will perceive objects and people in the other frames to be more massive - a perfectly matched, reciprocal symmetry. The upshot of these ruminations is that duality, symmetry breaking, and the transposition of field roles, must essentially be describing one and same phenomena. This equivalence is patently apparent in the limited, and much simpler case of the magnetic monopole. In string theory the existence of the magnetic monopole is seen as a duality of the electromagnetic field, but it also can be viewed as a transposition of field roles, while the great mass of the monopole is exactly what is expected from a symmetry breaking.

The Matrix Connection

To recapitulate, since the intermediate vector bosons - the W's and Z0 - appear at energies below the supersymmetry scale, the electromagnetic and weak forces are treated as one force in the field transposition conjecture. This reduces the number of transposable elements of the superforce to 3 - gravitational force, electroweak force, and strong force (QCD), which limits the number of possible permutations to six (N!, where N = 3, but there is an added complication that potentially greatly expands this number. But suffice it to say that the six permutations referred to here correspond to the lowest energy permutations). It was conjectured that these six (lowest energy) permutations, or field arrangements, corresponded to the six subtheories of M-Theory, and that the process of transposing roles of any pair, of the three fundamental forces, is equivalent to a supersymmetry type operation.

Furthermore, each of these six permutations results in the creation of a new pair of field structures that possess unique properties. To conform with the role exchange hypothesis each member of the pair must retain the fundamental structure of the original field from which it derived, but now couple to the other original field's type of force. This condition imposes a reciprocal trade of force strengths, so that in the new set of fields a previously weak force becomes strong, and a previously strong force becomes weak. But, it should be pointed out that in the case of transposing fields of equal strength, each new field ends up with the same original strength.

The intention of transposing roles among the three fundamental fields - gravity, electroweak, strong - inevitably poses a complication. In the simple case of the magnetic monopole one field (or charge type) swaps places with another - e.g. magnetic field (charge) substitutes for electric field (charge) role. For this strategy to work with the three fundamental fields, as defined here, they would need to possess the same number of charge types; technically known as degrees of freedom. Additionally, the definition of degrees of freedom needs to be broadened to accommodate gravity, whose components are not normally considered charge types.

Then, the question becomes which degree of freedom in one field transposes with which degree of freedom in another field? Clearly, a multiplicity of possible permutations obtains as a function of the number of degrees of freedom per field, so that the simple formula N!, where N=3 yielding 6 permutations applies to one particular subset and needs to be expanded. It can reasonably be assumed that the energy required to effect a particular field transposition will vary from one transposition to another, just as for example, it requires more energy to create a magnetic monopole than an antiproton. Thus for our purposes here, the six lowest energy permutations will be labeled major permutations to distinguish them from the remaining higher energy permutations that will be designated minor permutations.

The multiplicity of minor permutations is proposed to be the basis of the many possible solutions for each of the six subtheories of M-Theory. Since every permutation, major or minor, leads to a pair of fields with unique properties it is clear that a given transposition has two distinct solutions - the original set of fields, and the new set. Thus, if all possible permutations were arranged as a matrix table, the various elements would be non-commuting. Intriguingly, a recent analysis by Tom Banks of Rutgers University, and his colleagues, indicates that M-Theory can be described in terms of matrix theory involving non-commutative geometry. However, their analysis derives from the much deeper, more fundamental level of string geometry.

Degrees of Freedom, or the 333 Fold Way

By invoking several key assumptions, the number of degrees of freedom for each of the three fundamental forces can be equalized. Specifically, with the adoption of these assumptions, each force is described by a triplet of primary degrees of freedom, and a triplet of matching secondary degrees of freedom, that relate to the primary degrees of freedom by a local gauge symmetry. Thus, broadly speaking, each of the three forces exhibits the same overall structure, despite the great disparity in the nature of each force.


For the strong force (QCD) there are three types of color charge - red, green, blue - as denoted by its Lie group representation SU(3). The fields associated with these charges are designated as chromoelectric fields. However, these charge types occur also in anticolor versions - antired, antigreen, and antiblue, which play an integral role in gluon exchanges between quarks, and thus are normally counted as degrees of freedom, for a total of six. But, in addition to color and anticolor, gluons possess chromomagnetic fields analogous to the magnetic field of electromagnetism.

The chromoelectric and chromomagnetic fields of QCD relate to one another via a local gauge symmetry in a manner similar to the local gauge symmetry exhibited between electric and magnetic fields, and manifested by the photon. The major differences are that there are three types of color charge, which are carried by the gluons, drastically altering their behavior vis-a-vis the photon. Since the chromomagnetic fields relate to the chromoelectric fields through a local gauge symmetry, these chromomagnetic fields will be assigned here as the secondary degrees of freedom for the strong force, while the three types of chromoelectric field are assigned the status of primary degrees of freedom.


Remarkably, the electroweak field is also characterized by three primary degrees of freedom - namely, two types of weak charge, and one type of electric charge, as embodied indirectly in its Lie group representation - SU(2) × U(1). For the electric charge the secondary degree of freedom is, of course, the familiar magnetic field. But what about the two weak charges? In the standard model the weak force is carried by three intermediate vector bosons - the W+, W-, and Z0, but only two types of charge are involved. The first of these, the weak W charge, is carried only by left-handed electrons, which results primarily in the emission of left handed electrons in beta decay. The other one, the Z charge; carrier of the Z force, has opposite signs for left and right handed electrons. These two weak charges are sometimes labeled with the colors orange and purple, respectively, to distinguish them from the the three color charges of QCD.

To account for the secondary degrees of freedom in the weak charges, a slight modification to the standard model is introduced. It's postulated that each of the two weak charges have associated magnetic fields whose range and strength mirror that of the weak force. These putative weak magnetic fields would thus range only to about 10-16 centimeters. Because of the short range of these weak magnetic fields it's speculated that the transposition of weak electric and weak magnetic field roles can occur at much lower energies than the unification scale (1015 GeV) associated with Dirac monopoles. Such a transposition would, of course, give rise to weak magnetic monopoles - a north/south set for each type of weak charge.

These two types of weak monopoles are identified as the 2nd and 3rd generation neutrinos - muon and tau neutrinos - of the standard model. It's further proposed that the brief physical association of these weak magnetic monopoles, with the first generation leptons and quarks, gives rise to the existence of the two higher generations of fundamental particles (this overall idea is the subject of another paper by the author, and will not be elaborated on here). Thus, with the assumption of additional weak magnetic fields, the electroweak force would also possess three secondary degrees of freedom that each relate to their corresponding primary degrees of freedom via a local gauge invariance.


Gravity, at first glance, would appear to be the odd man out. How might it be embraced within the framework outlined so far? That is, how do you squeeze three primary, and three secondary degrees of freedom out of this force? The key to understanding this is the realization that the force of gravity arises, in general relativity (GR), from the warping of the space-time fabric in the presence of mass. In GR space and time form a union that possesses a Lorentz invariance, which in fact, is another form of local gauge invariance. If we assign space (or length scale) to be a primary degree of freedom, then time automatically acquires the role of a secondary degree of freedom for the gravitational field. But this still leaves us short by two primary and two secondary degrees of freedom for this field. Fortuitously, here is where mass enters the picture.

In GR the existence of space-time is completely dependent on mass. Without mass there would be no space-time; a position held by Albert Einstein himself. But where does mass come from? The Standard Model, to be internally consistent, requires the existence of a Higgs field that would permeate all of space-time. The masses of all the leptons and quarks, and the intermediate vector bosons - in effect, all mass are believed to derive from this field. In the simplest model of the Higgs field there exists only one electrically neutral Higgs particle. More general models allow for electrically charged Higgs particles, as well. The most widely accepted model calls for a trio of Higgs particles - one neutral, one positively charged, and one negatively charged.

Intriguingly, this Higgs field structure parallels that of the weak field which is mediated by a similar trio of particles - the W+, W-, and Z0. If we assume that this implies that the Higgs field is also characterized by two distinct charge types, then the triplet of primary degrees of freedom for the gravitational field is realized - space, and two Higgs charges. And, assuming these Higgs charges are accompanied by associated magnetic fields, as proposed for the the weak W and Z charges, there would be a corresponding triplet of, locally gauge related, secondary degrees of freedom. Thus mass, in an indirect way, provides for the two extra primary and secondary degrees of freedom in the gravitational field.

In summary, for each of the three forces of nature - strong, electroweak, gravity - as defined at and above the energy scale of the electroweak synthesis, there would be a triplet of primary degrees of freedom, and a triplet of secondary degrees of freedom. This gives an aesthetically appealing 3-3-3 × 2 structure of primary and secondary degrees of freedom, that relate to each other via local gauge symmetries. Or, conversely, each field can be viewed as possessing 6 total degrees of freedom for a 6-6-6 symmetry. Equalizing the degrees of freedom for all three forces rationalizes the transposition of field roles, for both major and minor permutations.

Supergravity and Quantum Mechanics

Knowing that each of the three forces - strong, electroweak, gravity - possess the identical number of degrees of freedom allows us to logically conceptualize the consequences of transposing the roles of any two of them. With the number of transposable elements in the above set equal to three, there are six possible permutations (N!, where N=3). Recall that it was conjectured that these six uniquely identified major field arrangements corresponded to the six subtheories of M-Theory, and that the process of transposing any pair, of the three fundamental forces, is equivalent to a supersymmetry operation. Supergravity is part of the tableau of M-Theory, so one of the six permutations must lead to the supergravity field structure.

Since supergravity incorporates a range of particles that couple strongly to the gravitational field, at least one member of the pair of fields to be transposed must be the gravitational field. Actually, there are only two possibilities: 1) Gravity is transposed with QCD, and 2) Gravity is transposed with the electroweak field. In each of these permutations to conform with the role exchange hypothesis, each field must retain its fundamental structure and strength, but now couple to the other field's type of force.

Thus, in the first choice a chromodynamic field with 8 gluons type particles will now be seen to couple via a gravitational type force, but possess the identical strength and range as the strong force we are familiar with. And, once again we are confronted with a remarkable, serendipitous coincidence. The N = 8 formulation of supergravity (the most favored of the formulations) calls for eight gravitinos that couple strongly via the gravitational field and possess a spin of 3/2.

The spin presents a problem though. In our universe gluons are bosons and possess a spin of 1, not 3/2. But, once again, perspective could resolve the contradiction. Supersymmetry is a spin related symmetry. Therefore, just as mass is perceived to be different depending on which side of a supersymmetry divide the observer is on, perhaps spin also is a function of perspective. Thus, a person made of supergravity type matter would measure his gluons (our N = 8 formulation gravitinos) to have the normal spin associated with strong force carriers, namely 1, while our gluons would appear to him to possess a spin of 3/2, and couple strongly via his version of gravity. If this assumption is true it has deep implications for the meaning of the quantum phenomena of spin.

The second permutation - gravity transposes with the electroweak field - may be a principle factor in the strange, counterintuitive properties of quantum mechanics in our universe. Assuming supersymmetric transformations begin just above 50 GeV, the new set of fields and particles deriving from this particular transposition should constitute a component of the natural vacuum fluctuations. Consequently, these virtual fields and particles should directly alter the behavior of fundamental particles, much as the polarization of the vacuum modifies the observed value of the electron's charge and mass.

While there are numerous minor permutations possible within this major permutation, one that is of particular interest involves the transposition of the space-time degree of freedom - within the gravitational field, with the electro-magnetic degree of freedom - within the electroweak field. To flesh out this major tranposition, the two weak charges of the electroweak field would be required to transpose roles with the two postulated Higgs charges of the gravitational field. To keep the analysis simple, and to focus on what may be the really important aspects of this interchange, the effects of transposing the weak charges with the Higgs charges will be ignored, for now.

The transposition of the roles of the electro-magnetic field with the space-time field must, of course, lead to two new fields with unique properties as seen from our perspective. One of these fields would possess the structure and coupling strength of our familiar gravity field, but couple to the electric and magnetic components of Maxwell's electromagnetic field. In effect, this new field would constitute an electric-magnetic continuum - an analogue of our space-time continuum, but whose components would couple extremely weakly (by the square root of the ratio of the strengths of gravity and electromagnetism) to our electric and magnetic fields. Nonetheless, the coupling between this postulated electric-magnetic continuum, and our electromagnetic field would still be 1021 times stronger than the gravity coupling between two masses.

The other new field would possess the structure and coupling strength of the electromagnetic field, but couple to the components of our gravity field - space and time (again by the square root of the ratio of the intrinsic strengths of gravity and electromagnetism). This putative field, existing as a backdrop of vacuum fluctuations, may play a pivotal role in quantum mechanical behavior. As an analogue of our familiar electromagnetic field, but with components of space and time, it will hereafter be referred to as the spatiochronetic (SC) field (where chronetic is derived from the Greek word chronos - for time). It follows that the spatial and temporal components of the SC field are linked via Maxwell's laws. Spatial charges associated with the SC field, would constitute the analogue of electric charges in our universe. Therefore by default, temporal charges associated with this SC field would constitute the analogue of magnetic monopoles in our universe.

The photon of the SC field must be massive, or otherwise we would experience it on an everyday level, as with the electromagnetic field. This means that its intrinsic range is microscopic. But paradoxically, the influence of the SC field might, nonetheless, range across macroscopic distances by virtue of the fact that a derivative effect of this field is temporal in character, and all matter in our universe is subject to the influence of time. The mechanism by which virtual SC fields might link quantum particles temporally to one another is proposed to be the sync shift, or relativity of simultaneity.

The sync shift is part of the suite of Lorentz transformations associated with the motion of reference frames relative to one another in our universe, and is not the same as the familiar time dilation effect. All legitimate time travel schemes rely on it. In special relativity the sync shift has opposite sign for bodies that are approaching or receding, and it is a coaxial phenomena exclusively. By coaxial is meant that unlike the relativistic time dilation effect, which is independent of the angle of motion between reference frames, the sync shift applies to that component of motion between reference frames that is directly towards or away from each reference frame. The sync shift is a distance related phenomena and increases with increasing distance between frames.

To be continued...