Why Topos

Why would Topos Theory be helpful in Physics ?

In physics there are a number of unanswered questions… Many efforts have been made by scientists/physicists to be able to answer those questions. Efforts which have lead to different paths of inquiry.

In this web site I would like to focus on one particular path which unfortunately is not very trodden.

The guiding principle of this path is the idea that the adoption of Topos theory in certain areas of physics , specifically Quantum Logic and Quantum Gravity might help shed light on how to solve some of the unanswered questions that arise in Physics.

Unanswered questions and Topos Direction Isham 1999 Raptis 2001

  • What is a Physics Theory?
    • The formulation of Quantum Theory rests on certain “a priori” assumption which, when seen from a general prospective, seems utterly unjustified . By this “a priori” assumption I am referring to the uses of real and complex numbers as the fundamental mathematical language in terms of which the theory is formulated. (For example the use of continuum to describe space time etc). In order to avoid such an “a priori” unjustified conception, what is needed is an abstract generalization of a Physics Theory such that Quantum Mechanics is seen as a particular application of this general construct. Isham and Döring have shown that this abstraction can be achieved through Topos Theory in the following way: imagine we have a system S and we want to find a Physics Theory of S. First of all we need to decide what type of Physics Theory we want to define: either Classical or Quantum. Say we opt for a Classical Theory. It is then possible to construct a Topos t S whose definition will depend on S and on which theory type we have chosen (in this case Classical Theory). Moreover, to each system S it is associated a unique higher order language L(S), whose elements and relations between the elements are specified “a priori”, but whose variables will depend on the system S. Given this unique higher order language L(S) of the system S, a Physics Theory is identified with a representation of L(S) in the Topos t S. Platonically speaking, one can view a Physics Theory as a concrete realization, in the realm of a Topos, of an abstract “idea” in the realm of logic. Therefore, this view presupposes that at a fundamental level, what there is, are logical relations among elements, and a Physics Theory is nothing more than a representation of this relations as applied/projected to specific situations/systems. For a detailed analysis of the above ideas see the series of papers: Isham, Döring, I 2007, Isham, Döring, II 2007, Isham, Döring, III 2007, Isham, Döring, IV 2007 . The slides of a talk given by Döring can be found at Talk1, Talk2 A recent ente encyclopedia article can be found at What is a thing?
  • How do you decide whether a statement is true or false in Quantum Mechanics ?
  • How can spacetime be described such that one doesn’t encounter singularities and non renormalizable infinities?
    • It has been discussed in various papers Mallios, Raptis 2001, Mallios, Raptis 2001, Raptis 2001, Raptis, Zapatrin 2000, Raptis, Zapatrin 2001, Isham 1991 that the description of space time using set theoretical concepts seems to be rather questionable . In fact it would seem that most of the infinities and renormalization problems, which arise in the Classical Theory of Gravity, are due to the current description of spacetime. Instead, if Topos theoretics ideas are used to describe spacetime, the above mentioned problems might be overcome, since, roughly speaking, point in spacetime would be replaced by something more general i.e. Locales. For further information the reader should refer to Raptis 2001, Raptis 2000, and the papers mentioned above.
  • How can one make sense of the multitude of consistent sets of History Propositions in the Consistent History Approach to Quantum Gravity?
    • Before answering this question a brief outline of the Consistent History Formalism is required. Roughly speaking the basic elements in the Consistent History Formalism are Histories identified as a sequence of events at different times. These events are represented by a set of orthonormal projection operators each of which represents a specific property at a given time, therefore, a History is represented by a chain of such projectors. Within this Formalism probabilities are assigned to sets of Histories of a closed system rather than to individual events. These probabilities, though, can only be assigned to sets of histories which do not interfere among them. For more information see Fay Dowker and Adrian Kent 1995, Halliwell 1994, Erich Joos, Pro and cons of the Consistent Histories approach to Quantum Mechanics and references therein. We can now turn our attention to the opening question. In a series of papers Isham 1996, Isham 1995, Isham and Linden 1994, Isham 1994, using Topos theory Isham describes a logical framework in which the probabilities of the theory are interpreted in a context in which every consistent set of histories is taken into consideration. The logical framework that is proposed by Isham deals with second level propositions, i.e propositions of the form “a history h is true with probability p”:= < h,p >. In particular Isham identifies the stage of truth (context in which propositions acquire meaning) of second level propositions (ex < h,p >), with the subset W of Boolean algebras2 (generated by the set of histories propositions which contains the history proposition (h) to which we want to assign a certain probability) of the set UP3 of all History propositions . While the truth value of a given proposition, at a particular stage of truth W, is related to the collection of coarse grainings of W which contain the history proposition h and which are consistent. Precisely the truth value of a second level proposition will be identified as a sieve defined on some W.
  • How can one make sense of the state vector reduction process which is viewed as an instantaneous, discontinuous process in the Copenhagen interpretation of Quantum Mechanics?
    • Isham showed in his paper Isham 2005 that it is possible to interpret/ describe the very controversial process of State Vector reduction using Topos Theory. In particular, given the set of all of all finite strings of projector operators which forms a Monoid, the collection subsets of the Hilbert space on which this monoid acts on the left forms a Topos. Within this frame work, the process of State Vector reduction is identified with left ideals of the above mentioned Topos. The interested resder can find an explanatory summary of the results of Isham’s above mentioned paper in A brief summary of State-Vector Reduction the language of Topos Theory

1 assignment of a truth value

2 The elements of a Boolean algebra generated by a set of Histories Propositions C={a,b,c…...} are formed by the tensor sum of element of C. Note that the subset of Boolean algebras of the set UP which are generated through sets C are automatically consistent, but there exists Boolean algebras which are not generated in such way, therefore need general definition of consistency see Isham 1996, for detailed definition.

3 UP is identified with the orthoalgebra of History Propositions i.e its elements are orthogonal projection operators each of which corresponds to a given proposition (alternative).