What really is a continuous symmetry?

Discussion in 'Physics & Math' started by arfa brane, Oct 27, 2020.

  1. arfa brane call me arf Valued Senior Member

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    Mathematically a continuous symmetry is a property of a Lie group.

    So if time is a continuous 'space' which looks like a continuous symmetry of nature, what does that mean? Time is always mathematically represented as an element of the real numbers, i.e. having just one dimension.

    Simultaneous events have the same real number assigned to them, in a larger space (mathematically speaking).
     
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  3. QuarkHead Remedial Math Student Valued Senior Member

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    No it's not. The Lie groups describe continuous symmetries - a rather different matter

    It doesn't mean anything - time is not any sort of space at all. In geometry, it is a dimension

    Oh? \(\sqrt{2}\) is a Real number. How many seconds is that, do you think?
     
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  5. paddoboy Valued Senior Member

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    Bingo! The geometry of spacetime.
     
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  7. arfa brane call me arf Valued Senior Member

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    Please explain in your own way the difference between a description of a continuous symmetry and a property, in some Lie group. Any group will do.

    Or explain why a group doesn't have properties, instead it has descriptions, if you can't manage the above.
    So the real numbers aren't a space either? Or we just have a map, for no particular reason, from the domain of time to real numbers, I suppose.
     
  8. arfa brane call me arf Valued Senior Member

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    7,110
    Well, if I choose a unit of time that I just say is √2 seconds, what can a mathematician do about it?

    Or say I construct a simple pendulum with a period of √2 seconds; are you saying I can't? What if I say a second is √2 units of my choice of a period?
     
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  9. arfa brane call me arf Valued Senior Member

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    What I should have said was all continuous symmetries are Lie symmetries, as far as we know so far.

    But now it's all about what a property is; and again I ask, when is an object distinct from its properties? When is time distinct from an observed continuous, constant flow or "movement", the properties time appears to have? When is a photon distinct from its frequency? Etcetera and so on.
     
  10. Michael 345 New year. PRESENT is 70 years old Valued Senior Member

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    What properties would those be please?

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  11. arfa brane call me arf Valued Senior Member

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    But you probably didn't read it the first time.

    So, here goes again:

    Property 1) time is continuous
    Property 2) time flows at a constant (linear) rate, locally.
     
  12. Michael 345 New year. PRESENT is 70 years old Valued Senior Member

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    Makes no sense

    Property 1) (Insert WHICH property here) is continuous
    Property 2) (Insert WHICH property here) flows at a constant (linear) rate, locally

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  13. Write4U Valued Senior Member

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    At the spatial beginning of the universe the flow of simultaneously emergent time was created and has existed ever since, but only as a result of a continuing spatial symmetry.
     
  14. arfa brane call me arf Valued Senior Member

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    It does make sense. Which property of time is the one that is the continuous property? Likewise, which property of time is the constant flow?

    When is time distinct from either of these properties?
    Ergo time is these properties, that (we say) it "has". That's an incorrect thing to say, but we know that.
     
  15. Michael 345 New year. PRESENT is 70 years old Valued Senior Member

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    Please name as many properties of TIME as you can

    Thanks

    Definition of property

    1a: a quality or trait belonging and especially peculiar to an individual or thing

    b: an effect that an object has on another object or on the senses

    Merriam-Webster

    Note TIME is not a individual or thing
    also TIME is not object

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  16. QuarkHead Remedial Math Student Valued Senior Member

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    OK, try this tough guy. The group of all Real \(3 \times 3\) matrices with determinant \(+1\).

    This is a Lie group, known to its friends as \(SO(3)\). It has a representation as the vector space of all linear automorphisms on \(\mathbb{R}^3\) and, like any Lie group, it is a \(C^\infty\) manifold. This manifold has the topology of the Real 2-sphere.

    Any wiser?
     
  17. arfa brane call me arf Valued Senior Member

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    7,110
    Time is an individual thing. Time is not like distance; but it can be made to look like one.
    A distance is an object, a physical object. I can prove that mathematical objects exist, I can prove that it's physically possible to measure a distance or equivalently, compare it with a unit of distance.

    In physics, making time look like distance is a commonly used method of plotting a function of time, like a velocity.
     
  18. arfa brane call me arf Valued Senior Member

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    7,110
    Ok.

    Are the 3 x 3 real matrices with det 1, a representation of this group SO(3), are there other reps?; are the matrices properties of this group, or are they descriptions of continuous symmetries, or just mathematical objects? What's the group action? Is a particular notation for the action also a property or a description?

    Can this group be considered a set of rotations acting on an abstract vector in \( \mathbb R^3 \).

    p.s. when do I get an answer to my question, that explains the difference between a property (of a group) and a description (of a continuous symmetry)?
    I think the one that starts with "groups don't have properties" is just wrong, because they do.
     
    Last edited: Oct 29, 2020
  19. Ethernos 1997 Registered Senior Member

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    1st we decribe what 1 secound is then we write 1 secound =(√2 ×√2)-1 secound or √2=1.41421356237+ c¹¹
    c¹¹ is constant describing if u add 1.41421356237 i.e. value of √2 after 11 decimal no. and if added would give √2. hope it makes sense.
     
  20. Ethernos 1997 Registered Senior Member

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    or u can take time interval as √2 i.e. t sec=√2 sec. if u lived in√2 universe it would be faster than our universe.
     
  21. arfa brane call me arf Valued Senior Member

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    That sounds vaguely like some kind of warning. So, thanks; I for one won't be going there.
     
  22. arfa brane call me arf Valued Senior Member

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    --https://en.wikipedia.org/wiki/Orthogonal_group
    --ibid.
     
    Last edited: Oct 29, 2020
  23. arfa brane call me arf Valued Senior Member

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    7,110
    What I would list as some properties of a group:
    A group is a set with a binary operation acting on pairs of elements from the set. A set can have several different-looking representations, but these can be shown to be different notations for the same object (the group itself).
    A group has an identity operation, when this is paired with any other group element, the group operation leaves that element unchanged. The identity can act more than once, therefore, in fact it can act infinitely in a group with discrete or continuous symmetry.

    Rotations are equivalent to compositions of pairs of (not equal) reflections, in any orthogonal group. This is another property of rotation/reflection groups in general.

    According to: https://en.wikipedia.org/wiki/Circle_group#Properties

    Properties
    Every compact Lie group G of dimension > 0 has a subgroup isomorphic to the circle group. That means that, thinking in terms of symmetry, a compact symmetry group acting continuously can be expected to have one-parameter circle subgroups acting; the consequences in physical systems are seen for example at rotational invariance, and spontaneous symmetry breaking.

    The circle group has many subgroups, but its only proper closed subgroups consist of roots of unity: For each integer n > 0 , the nth roots of unity form a cyclic group of order n, which is unique up to isomorphism.

    . . . and as we should know, there are many ways to represent a permutation group or a cyclic subgroup of one. Also notice that nth roots of unity are 'fixed' in the complex plane, by inclusion in the set \( \mathbb C\).
     
    Last edited: Oct 29, 2020

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