String Theory from Logic Alone
It is my contention that physics can be derived from the principles of logic alone. I describe here how this may be accomplished. If you wish to contribute to this effort, please feel free to email me with your suggestions. The basic premise is that a mathematical description of logic is accomplished by the union and intersection of regions of sample space. These regions are also known as "events" in sample space. The boarder that defines an event is the surface that encloses a region of this space. Yet the region of an event can never be completely enclosed because samples are continually being added with time. The result can be seen as a circular loop that sweeps out more of space with time. This loop or closed string moving through time is also known as a world-sheet. The way these strings split apart and join together is basically the same as the interactions described by Superstring theory.
The laws of physics are invariant with changes in coordinate system. That is, the properties they described do not change when you change the coordinate system. But invariance with changes in coordinate systems requires the existence of a manifold. A manifold is a set of objects called a topology with properties that allow each point in the set to be associated with coordinates. A topology is a collection of subsets of some superset such that unions and intersections of these subsets are also included in the collection. Reality can be considered to be a topology since unions and intersections of any portions of reality are also included in reality. Unions and intersections are alternative expressions for the AND's and OR's of logic. So the invariance of physical laws prove the necessity of some underlying logic to construct anything real.
I put some effort into showing how the principles of logic can be represented graphically with objects in some arbitrary space (See A Graphical Representation of Logic). Then I show how a coordinate system can be imposed on this space and how logic can be described with functions inside regions of this space (See Probability Densities). Yet these efforts only justify the existence of a manifold used to represent logic. And I expanded on the types of logic that can be represented.
The sample space that I mention is just another way of describing a manifold where unions and intersections of regions are still included in our considerations. The surface that encloses a region is a submanifold of the entire space. This would be the language used in physics. By saying that even the most fundamental constituents of reality are constructed of manifolds (sample space) is a recognition of the fact that we don't know at what level matter stops being constructed of even more fundamental objects. There may always be smaller things of which larger things are constructed, even to the infinitesimal. If so, then unions and intersections of these smaller objects are still included within the larger ones. And there again we have a topology that we are trying to mathematically describe so that we must have a manifold. You can't justify properties of an infinitesimal point. For there is nothing inherent inside a singular point whose variations or interactions can explain those properties. So the properties of even the most tiniest bits of matter can only be explained if it consists of a topology of smaller objects that can each be assigned coordinates. Thus, even the smallest of objects must be a manifold. Not only that, but if the universe started from a singularity, then this proves that everything in reality reduces to nothing more than infinitesimal points in space. All events are just collections of points in space. So now we have no alternative but to consider how collections of points (events in sample space) grow with time and interact.
I do not have a PhD in physics or math. I've only noticed an alternative perspective that may provide a more complete explanation for the laws of physics. Whether this is actually equivalent to String Theory is complicated by the fact that String Theory has not yet been reduced to the simplest axioms, and I am trying to find similarities in fundamental concepts. I can only hope that those who are experts in these areas of study will consider this perspective.
Consider...
Normally, an event is represented by some region in sample space as shown in Fig 1. Certainly the whole universe can be considered to consist of samples of things contained within the one big event of the universe. But each sample itself occupies a small region of sample space. So each particle can be considered to be a sample that occupies a region of samples space. And it is hoped that the study of the geometry of this situation may result in quantities that can be interpreted as physical properties such as the mass and spin and the charge of particles, etc.
But not all the events to occur have happened yet and cannot be added as samples to the universe as a whole. Each time we take a measurement or observe an effect adds one more sample to our considerations. So the region that represents an events cannot be a completely enclosed region. We cannot enclose all of reality into one single event because things continually happen with time. Time is the result of open events. Neither the universe as a whole nor the particles themselves can be represented by closed events because it changes with time. So we must study the geometry of open events as shown in Fig 2.
Things change with time, meaning particles exist and interact, as a result of the event being open. It is the open part of the surface of an event in which particles and interactions are defined. Thus, physicists use things such as Feynman integrals, action integrals, and Euler-Lagrange equations in an attempt to close this open region with a parameterization of a surface as shown by S1 in Fig 3. The parameters used to close these events are the usual parameters of time found in the classical study of physical field theories. Classical physics is described by the best surface that can close the event.
Fig 3 shows S1 completely closing the region as a representation of events at one instant of time. But the manifold of reality cannot be closed at the end for the same reason that it cannot be closed at the beginning - because time is still changing things. And so Fig 4 illustrates time changing the overall event from one boundary B1 to another B2.
These boundaries are similar to the closed strings of Superstring Theory. And since every surface between these two boundaries is just as much a valid possibility as another, we need to add up all the possible contributions from every possible surface that can be constructed from ST1 to ST2 as shown in Fig 5. This is the Feynman formulation of quantum mechanics which gives the particle characteristics. Here, each surface from ST1 to ST2 is a "world-sheet".
According to Brian Hatfield's book, Quantum Field Theory of Point Particles And Strings, page 481, "...string theory will find an application any time a problem can be formulated as a sum over random surfaces... The sum over all world-sheets is a sum over random surfaces, hence any sum over surfaces can be interpreted as the propagation of some string." This is what we are dealing with here.
So the question is, if the fundamental particles of matter are really strings that sweep out more (sample) space with time, what characteristic values can particles have, how are these values calculated, and what interactions can exist between particles?
Below I show a number of different ways a string might propagate as it sweeps out more space with time. Every possible interaction must start with strings sweeping out more space and end with strings sweeping out more space, for that is how we have defined events growing with time. Fig 6 shows no interaction at all as a string moves through the dimension of time.
Fig 7 shows two string particles, A and B, interacting by joining together to form another string C. The strings actually join at time to. It is also possible that one string might split into two separate strings as shown in Fig 8.
Another possibility is that two strings might join momentarily and then split apart again as shown below in Fig 9. At time t0 we start with two strings A and B, then at time t1 there is only one string, and then at time t2 there are again two strings C and D. It may be possible that C has all the characteristics as A, and D is the same a B. Or D may be the same as A with C the same as B. Or C and D may be entirely different from either A or B. Notice that Fig 9 is a combination of Fig 7 joined on the right by Fig 8. This might be interpreted as two strings bounding off each other. And then again, another possible interaction of strings may be where one string splits apart and then rejoins as shown in Fig 10. There is only one string at t0, then two strings at t1, then one again at t2. And notice here that Fig 10 is a combination of Fig 8 joined on the right by Fig 7.
And there are many other possible interactions between strings that could be drawn. But these would just be higher order combinations of the above and could get very complicated indeed. Yet, in each case you start with strings and end with strings, and the number of strings at the end need not be the same as what you started with.
So we have the possible ways that string particles can interact, but how does that help us determine the characteristics of strings? Certainly, if all the particles of nature passed right through each other and no particle ever interacted with another, then nothing would interact with nothing, there would be no effect from a given cause, and we could learn nothing about particles or about nature at all. The characteristics of particles can only be revealed if particles interact with each other. Whatever characteristics a particle may have that do not pertain to how it interacts with other particles is by definition of no consequence to other particles nor to us. The only relevant particle characteristics are those that give information on how likely interactions are to occur.
In the average of very many interactions over long periods of time, these interactions will look like a force acting on a charge such as gravitation pulling on a mass or an electric field acting on an electron. But on the scale of just a few particles interacting, force and charge are a measure of how probable interactions are. The more likely it is for a string to split apart and join together, then the more interactions are likely to occur. The more likely interactions are, the stronger the force, since each string is affected more often. And the probability of interaction will depend on the number of ways that the particles can interact.
Or, if all the characteristics of particles could arbitrarily change with time of their own accord, without the need of interactions, then we could not distinguish effects caused by interactions from arbitrary effects. Again we could not distinguish cause and effect, and there would be no predictable pattern to reality. So we are looking for characteristics that do not change with time; we are looking for characteristics that only change as a result of interactions with other particles. These are called conserved quantities. And the question is whether we can find such conserved quantities in this scenario of growing events.
More to come...
And so this scenario raises some questions such as what process is responsible for this manifold of reality to begin with? Why do we perceive only 3 dimensions? And what are the consequences of an ever increasing manifold?
To start with, a critical point is where df=0, all partial derivatives of a function (here the probability density function) with respect to any coordinate system are zero. And so the dimensionality of the space at that point cannot be specified, and no logic can be described. Such a point certainly exists at a point where the probability density function and all its derivatives are zero, in other words, where everything is zero. But such a point is disallowed in any boundary or manifold because it identifies a point where things cannot be specified. So if some manifold exists, as it must if any mathematical description of logic is at all possible, then such a critical point must be removed from any manifold and thus leaves a hole in whatever manifold exists as shown by A in Fig 6. So total none-existence itself can be considered to be a critical point that then creates an open manifold from which time must flow.
But why do we perceive only 3 dimensions? Whatever dimensionality the manifold of reality has, a region of this sample space is described by a hypersurface that is itself a manifold of one dimension less then the original space. In 3-D a region is described by a surface that is a manifold of 2 dimensions. So in a 4 dimensional space, a hypersurface would be a 3 dimensional manifold. In an n-dimensional space, a region would be described by an (n-1) hypersurface.
Now, if the hypersurface is an open manifold, not completely enclosing a region, then the boundary of the hypersurface is itself a manifold of one less dimensionality. For example, in 3-D the surface is 2-D, and the boundary of an open 2-D surface is a 1-D manifold, a curve shown as B in the Fig 6. In an n-dimensional space, hypersurfaces are (n-1) dimensional manifolds, and boundaries of these hypersurfaces are (n-2) "hypercurves".
So generally, there is a 3 dimensional spread between the space, the hypersurface, and the boundary to that surface. Whatever the dimensionality of the sample space, 3 dimensions are required to represent a region of sample space prevented from being completely closed by a continuous edge. It may be that there are enumerable or even undeterminable dimensions in reality, but we perceive only 3 dimensions because that is all that is necessary to describe the evolution of events. If for example reality consists of 10 dimensions, the hypersurface would be 9 dimensional, and the edge of that hypersurface would be 8 dimensional. But the edge, B of Fig 6, may appear to us as a one dimensional strings because lower dimensions do not enter the calculation. They may appear as constants or have little effect.
Since time is forever pushing the boundary of events in order to include more samples with time, could this ever increasing boundary be responsible for entropy? For if we must always add more samples with time, then the probability for events can only increase with time. This might also be responsible for the expansion of the universe. If the boundary of the universal event increases, then space would expand with it.
Time in this scenario is not a true dimension. For we can arbitrarily move in any direction of space, but we cannot arbitrarily move through time. We cannot go back through the dimension of time, and we cannot go more forward through time than time will permit. We are simply stuck here only feeling the effects of time. And there is no degree of freedom from being pushed along by time. It is only when we consider history that we must specify a moment of time along with a place. We consider time as completely independent from space and thus as a different dimension.
My description here has yet to prove to be the same as the Superstring Theory studied in modern physics. The challenge is to formulate this description of open events in sample space in terms of modern string theory. It is hoped that this formulation can account for the number of dimensions and particle characteristics calculated by modern Superstring Theory. If this can be done, then we will have a theory of creation that derives from principle alone. And we would finally know exactly WHY physics is as it is.
