# What really is a continuous symmetry?

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

1. ### arfa branecall me arfValued 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. ### QuarkHeadRemedial Math StudentValued 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. ### paddoboyValued Senior Member

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Bingo! The geometry of spacetime.

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7. ### arfa branecall me arfValued 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 branecall me arfValued Senior Member

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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?

Ethernos 1997 likes this.
9. ### arfa branecall me arfValued 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 345New year. PRESENT is 71 years oldValued Senior Member

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What properties would those be please?

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11. ### arfa branecall me arfValued 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 345New year. PRESENT is 71 years oldValued 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. ### Write4UValued 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 branecall me arfValued 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 345New year. PRESENT is 71 years oldValued 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. ### QuarkHeadRemedial Math StudentValued 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 branecall me arfValued Senior Member

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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 branecall me arfValued Senior Member

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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 1997Registered 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 1997Registered 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 branecall me arfValued 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 branecall me arfValued Senior Member

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--https://en.wikipedia.org/wiki/Orthogonal_group
--ibid.

Last edited: Oct 29, 2020
23. ### arfa branecall me arfValued Senior Member

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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