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Rigorous definitions in quantum mechanics

  1. Jan 27, 2013 #1

    It is question of specifying mathematical definitions which are cummunes in several theories. In classical physics, in special relativity, in quantum theories (wave functions and state vectors) and in general relativity, we can assert :

    (a) The universe U is a topological space whose elements are called events and as each event has a neighborhood homeomorphic to R^4.

    (b) A local coordinate system is a homeomorphism between an open subset of U and a bounded subset of R.

    (c) A world line segment is a continuous function which is defined on an open subset of R and takes values in U.

    (d) A generalized physical space (a set of spatial positions) is a particular family of world lines of material bodies. For example, in general relativity, a generalized physical space of Rindler consists of world lines of a family of Rindler observers. http://en.wikipedia.org/wiki/Rindler_coordinates#The_Rindler_observers

    (e) To define a time variable in a generalized physical space we just have to choose a particular parametrization along each of his world lines or (in a corpuscular model) we just have to choose a particular parametrization along the world line of the body whose movement is studied. For example, a Poincaré-Einstein dating carried out by an experimenter P is a temporal variable (t) and a Poincaré-Einstein dating carried out by an experimenter P' is another temporal variable (t'). A Poincaré-Einstein dating carried out by an experimenter P is a temporal variable obtained by this method : the date associated with an event A is the arithmetic mean of the dates of issuance and receipt by P of a light signal which is reflected in A.

    (f) A physical space is a particular family of world lines of material bodies that is associated to a unique observer. Each of these world lines seems to him continuously immobile. We use a physical space to define a wave function in quantum mechanics, we use it to define the motion of a body in a corpuscular model, we use it to define the Doppler effect (which results from the motion of the source in the physical space of the receiver).

    The (f) is used in all other theories but is not yet specified in general relativity.See observational frames of reference : http://en.wikipedia.org/wiki/Frame_of_reference

    A theory which defines all the physical spaces of nature (some with regard to the others) is compatible with all the quantum mechanics.

    (1) In classical kinematics we postulate the existence of a universal chronology and we suppose that all the experimenters notice the same spatial distances between pairs of events that are simultaneous with respect to this chronology. We establish then strictly the possible states of movement between two physical spaces R and R'. We obtain that R'can be uniform or not niform translation with respect to R, this movement being coupled to a uniform or not uniform rotation.

    (2) In a relativist theory where there are several possible chronologies (for example, the datings of Poincaré-Einstein realized by diverse observers) and where none of them is favored, it is still necessary to specify the possible states of the movement between two physical spaces R and R'.

    Best regards.
    Rommel Nana Dutchou
    Last edited: Jan 27, 2013
  2. jcsd
  3. Jan 29, 2013 #2

    Do you agree these assertions :

    * A physical space is a particular family of world lines which are continuously immobile according to a unique observer. We have to express the use of a physical space to write a wave function in quantum mechanics.

    * There are several physical spaces in the nature and their definition (the states of movement of some with regard to the others) is a problem of kinematics.

    * To define a temporal variable in a physical space it is enough to choose a particular parametrization along each of its points (world lines). It is necessary to define a temporal variable to write a state vector in quantum mechanics.

    Thank you.
    Rommel Nana Dutchou
  4. Jan 29, 2013 #3


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    Welcome to PhysicsForums, Rommel!

    Do you have a particular point to make, or a question to ask? That would be more likely to prompt additional discussion.
  5. Jan 29, 2013 #4
    Thank you

    I just wanted to present this point of view and see if it is universal or not.
  6. Jan 29, 2013 #5


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    Since you are new to this forum, we know nothing about your background. Please start by telling us which textbooks and published papers on quantum mechanics, quantum field theory, and general relativity you have studied, and what level of education you have reached. Then, for each textbook or published paper you mention, tell us whether you believe they align with your point of view, or not, and how.
  7. Jan 30, 2013 #6
    Hello strangerep
    I made university studies in mathematics, completed by a training in financial mathematics. But I am interested for a very long time in the kinematics in physics, with the aim of building a theory which contains the special relativity and which specifies the notion of observational frame of reference.

    When I speak about quantum mechanics, I am only interested in the mathematical definition of a wave function and a state vector. I think that a set of spatial positions used to write a wave function has to be a particular family of world lines associated with a unique observer because we can write the variation in time of the probability to find a particle in one region of the space. He has to exist several physical spaces in quantum mechanics, each being associated with a unique observer.

    I drafted a document which defined the notion of observational frame of reference and I can realize the originality of my work by consulting this page : http://en.wikipedia.org/wiki/Frame_of_reference

    Thank you.
  8. Jan 30, 2013 #7


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    Umm... so... did you also study relativity at university, and/or by self-study from textbooks? (If so, which ones?)

    In your university mathematics courses, did you study: (1) linear algebra, (2) group theory, (3) functional analysis?

    If you studied functional analysis then you probably know what a "Hilbert space" is.
    A state vector is a vector in an abstract Hilbert space.

    If you don't know what a Hilbert space is, then you definitely need to study a textbook -- such as Ballentine's "Quantum Mechanics -- A Modern Development". The 1st chapter therein covers the mathematical tools quite well, at least for someone who has already studied some university-level math.

    The transformations between observers in special relativity (or indeed in non-relativistic Galilean mechanics) play the role of relating one observer's measurements to another's. I'm not sure that's what you meant though.

    (If you don't know what I'm talking about in the previous paragraph, then you also need a textbook on special relativity. For this, I like Wolfgang Rindler's books.)
    Last edited: Jan 30, 2013
  9. Jan 31, 2013 #8
    Dear strangerep

    Thank you for worrying you about my knowledge. But it is not for it that I opened this discussion. I have write precise definitions and I wonder if they are mathematically rigorous for everybody. Their physical interest has no importance in this discussion.

    Thank you again !
    Rommel Nana Dutchou
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