cedric flamant. outline what is symmetry? what is symmetry, to a physicist? physical symmetries uses...

Cedric FlamantSymmetry In Physics

Cedric FlamantSymmetry In Physics

Outline• What is Symmetry?• What is Symmetry, to a Physicist?• Physical Symmetries• Uses of Symmetries

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What is Symmetry?• Many physics articles nowadays mention symmetry

SupersymmetrySymmetry BreakingCP symmetry

Symmetry GroupsGauge SymmetriesSymmetry Transformations

• And for good reason!“It is only slightly overstating the case to say that

physics is the study of symmetry.” - Philip Anderson, Nobel Laureate (1977)

• But, the connection isn’t so obvious!

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What is Symmetry--to physicists?

Physical Symmetries-- Rotational-- Translational-- Time-- Lorentz-- C-symmetry-- P-symmetry-- T-symmetry-- CP-symmetry

Uses of Symmetry-- Constraints-- Noether’s Theorem

Conclusion

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What is Symmetry?• We have an idea of what symmetry means in everyday life

Sphere Strawberry

• Radial Symmetry• Mirror symmetry

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What is Symmetry?• Symmetry – From Greek συμμετρία, symmetria,• “agreement in dimensions, due proportion, arrangement”

Celtic knot Taj Mahal

• In physics, we need a mathematical definition for the concept to work with equations

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What is Symmetry, to Physicists?Symmetry – the symmetry of a physical system is defined as the invariance of some physical or mathematical feature under some transformation.

• Invariance of a feature = the feature remains unchanged

• In the words of Mathematician Hermann Weyl,“A thing is symmetrical if there is something you can do to it so that after you have finished doing it, it looks the same as before.”

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• We can easily relate this definition to the symmetry we are used to just by • Identifying the invariant feature• Identifying the transformation

Invariant feature – its shape, the region it fills up

Transformation(s) – 90 degree rotations, flipping along certain axes

Square

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Transformation(s) – 90 degree rotations, flipping along certain axes

Square

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Transformation(s) – rotation about any axis passing through the middle, by any amount.

Sphere

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• Symmetry to physicists is more general than just geometric notions!

Invariant feature – Social Security Number

Transformation – Switching to a new job

Social Security Number is symmetric under change of employment -

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Invariant feature – Your NameTransformation – Switching to a new job

Legal name is symmetric under change of employment -

But, it might not be symmetric under marriage!Feature – Your Name Might not be invariant!Transformation – Getting married

• Symmetry to physicists is more general than just geometric notions!

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Physical Symmetries• Imagine empty space, devoid of anything except you,

floating in a space suitOutline




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Physical Symmetries• Your thrusters malfunction and go off, spinning you every

which way for minutesOutline




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Physical Symmetries• Your suit eventually autostabilizes you, bringing the

spinning to a halt

• What direction are you facing? Is it the same as before? Would you be able to tell?

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Physical Symmetries• It’s not just you, no known physical process can tell the

difference between the different directions of space.• We can’t define an “absolute direction” to orient space.

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Physical Symmetries – Rotational• Only relative orientations of objects matter, not orientation

relative to space.

• If time paused and all the objects in the universe were rotated together, then time unpaused, no one would ever know the difference.

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Physical Symmetries – Rotational

• No experiment would be able to inform us of the rotation transformation.

• The stars would still be in the same place in the sky,• Canada would still be north of the United States,• Interactions between atoms would still obey the same laws.

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Physical Symmetries – Rotational• This independence from an absolute orientation is what we

would expect of physical laws from experience.

• Regardless of what experimental setup, whether it’s an electron diffraction experiment, or a clinical trial of a new drug, we expect that the direction of the setup is immaterial – the same processes occur in any direction.

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Physical Symmetries – Rotational• Aha, but what about a compass? The needle swings a

different amount depending on your orientation relative to the north pole!

• In such a case, the earth is also part of your physical system. The compass’s orientation relative to earth is important to the physics, but not relative to space.

• If you rotated both the earth and your compass together and redid the experiment, it would come out the same way.

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Physical Symmetries – Rotational• We conclude that the laws of physics are rotationally

invariant, and exhibit rotational symmetry. That is,

Invariant feature – Any given law of physicsTransformation – rotation about any axis, by any amount

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Physical Symmetries – Translational• Similarly, the laws of physics are translationally invariant,

hence obeying translational symmetry.

Invariant feature – Any given law of physicsTransformation – translation in any direction

Shift over

So, if time were to pause and all the objects in the universe were to be shifted over in space together, no one would know.

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Physical Symmetries – Translational• Once again, this is what we would expect from experience.• We expect an experiment conducted in Pasadena to

behave the same way as it would in New York, or in Ouarzazate – all thanks to translational symmetry.

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Physical Symmetries – Translational• Mathematically, a translation transformation is applied to

an equation through the rule . Wherever the position appears, we replace it with the shifted one.• Consider applying this transformation to Coulomb’s law in

electrostatics:

We get back the same equation, so Coulomb’s law has translational symmetry.

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Physical Symmetries – Time-Translation

• There is also time translation symmetry. Physical laws respect this symmetry as well.• We imagine this transformation as pausing a system,

shifting it to a different time, and restarting it. • A vase tipped in exactly the same way tomorrow would

break the same way as today.

Today Tomorrow

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Physical Symmetries – Time-Translation

• The time translation invariance of physical laws is particularly important – in fact, all of science depends on it!• This invariance allows us to assume that the physics of

today will be the same as the physics of tomorrow, or the physics of last century, or even the physics of billions of years ago.• If the laws of physics changed with time in some arbitrary

way, there would be no point studying physics, or any other science, or engineering for that matter.

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Physical Symmetries – Lorentz Invariance

• Physicists are also well-versed in less familiar symmetries, but the idea is no different from the ones we’ve seen.• There are quantities that are invariant under Lorentz

transformations – this transformation allows us to go from one frame of reference to another.• For example, it would tell us how an experiment on earth

looks like from the reference frame of a man on a spaceship speeding past.

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Physical Symmetries – C-Symmetry• C-Symmetry is the symmetry of some physical laws under a

charge-conjugation transformation.• That is, you take every positive charge and replace it with a

negative one.• In our Coulomb’s law formula, this corresponds to the

substitutions and .

• Note that the formula stays the same – hence Coulomb’s law is C-symmetric. In fact, electromagnetism, gravity, and the strong force obey C-symmetry, but the weak force does not.

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Physical Symmetries – P-Symmetry• P-Symmetry is the symmetry of parity transformation, the

change of the algebraic sign of the coordinate system:

• It’s reflecting the world through a point, changing a right-handed coordinate system to a left one.• For a while we suspected that nature would not care about

the difference between left and right – but in 1957 C. S. Wu confirmed that the beta decay of Cobalt-60 violates P-Symmetry.• Once again, it’s the weak interaction that messes with the

symmetry – the other three fundamental forces obey it.

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Physical Symmetries – T-Symmetry• T-Symmetry is the symmetry of physical laws under a time

reversal transformation.• There are some physical processes that still behave like real

physical systems when time is run in reverse – think of it like playing a video of the phenomenon in reverse.• A thrown ball follows a parabolic trajectory in the same

way, whether time runs forward or backward.

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Physical Symmetries – CP-Symmetry• The previous three symmetries can be combined together

– CP-symmetry describes the symmetry under both the charge conjugation and parity transformations.• When P-symmetry was found to be violated by the weak

interaction, CP-symmetry was still thought to be fundamental for all physical processes. • But, in 1964 James Cronin and Val Fitch provided clear

evidence that CP-symmetry could be broken – once again, the weak interaction was the culprit.

• This is what we mean by CP violation.

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Uses of Symmetry – Constraints• So far symmetry in physics has proven to be a nice way to categorize

things, but it is so much more!• Symmetry can be used to constrain theories, for example.• If you were asked to draw a curve that ends where it starts, you

might draw any of these:

• There are plenty of curves you can draw! But, if you’re told that the curve must be symmetric under any rotation around a specific point, you can only draw this:

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Uses of Symmetry – Constraints• Imagine we are interested in an unknown force acting

between two particles in empty space.

• The force might depend on various parameters of the particles, including their positions• Each particle has an x,y,z location to it, so in total just the

position introduces 6 parameters.

1

2

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Uses of Symmetry – Constraints• Let the positions of the particles be given by and

• However, if we were to shift these particles somewhere else in space, we would expect the force between them to be the same (translation invariance).

• This means the position fixing the system in space, say, , is irrelevant. Only the relative position between the two can matter: -

1

2

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Uses of Symmetry – Constraints• So, now we are down to only one relevant vector, - . 3 parameters!

• Let us also invoke the rotational invariance of physical laws. If we were to rotate the whole system in empty space, it should not change the physics either.

• So, consider rotating the system about the position of particle 1.

1

2

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Uses of Symmetry – Constraints

1

2

1

2

1

2

• All of the above rotations of particle 2 around particle 1 have the same physics, and hence the same force dependence on position of the particles. • Thus, the direction of one particle relative to the other does not

matter either! Only the distance between the two particles does.• This leaves us with the dependence on .

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Uses of Symmetry – Constraints• So, the force can only be a function of , the

distance between the two particles. We went from 6 parameters to 1!

• Does this dependence look familiar?

1

2

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Uses of Symmetry – Noether’s Theorem

• Emmy Noether proved an astounding mathematical result relating symmetries and conserved quantities.• In essence, if a system has a continuous symmetry property, then

there are corresponding quantities whose values are conserved in time.

• Continuous symmetries are associated with transformations that can be arbitrarily close to doing nothing to the system:• Spatial translation,• Spatial rotation,• Time translation.

• Not continuous (discrete)• Time reversal,• Charge conjugation,• Parity transformation

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• What are conservation laws?• The familiar linear momentum, for example!

• You are likely to have heard of or even used conservation of momentum to solve problems in your physics or engineering classes. • If you have used it, you know how useful it can be!

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• It turns out that the conservation of linear momentum is implied by the translational invariance of physical laws!• Just the fact that physics does not care whether you do an

experiment right in front of you, a millimeter to the left of you, or in Jakarta, mathematically implies conservation of linear momentum!• It’s a truly remarkable result

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Symmetry Invariance Conservation LawTranslation Symmetry Conservation of Linear Momentum

Rotation Invariance Conservation of Angular Momentum

Time Translation Invariance Conservation of Energy

Gauge Invariance Conservation of Electric Charge

SU(3) Gauge Invariance Conservation of Color Charge

SU(2)L Gauge Invariance Conservation of Weak Isospin

Probability Invariance Conservation of probability

Now we start to see why physicists are so excited about symmetry! Discover a new symmetry, get to make use of a new conservation law!

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• Noether’s theorem also provides a neat little explanation for “toy” problems you might have encountered at school, where for some reason, momentum or energy was not conserved.

• Imagine a perfectly elastic ball of mass m bouncing off a wall – it goes in with velocity v, and comes out with velocity -v

pinitial = mv

pfinal = -mv

• The change in momentum of the system is pfinal – pinitial = -2mv, but by conservation of momentum it should be 0!

• What happened?

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• You could say that the wall was not considered as part of the system, even though it is, so you can’t do a momentum balance

• But what if you do consider it to be part of the system? You could think of it as infinitely massive, also effectively justifying the wrecking of the momentum balance with some good ol’ infinities… but the wall is not moving?

• “What a disaster,” you exclaim triumphantly

• You could say, “well, the brick wall is actually connected to the earth, so the earth recoils, but since it’s so massive you don’t really see anything move.”

• Such a limiting procedure is correct, but you end up ruining the “toy” problem by dragging reality into it. That’s no fun, we should be able to do physics on models too.

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• It turns out by introducing a fixed brick wall, which cannot be moved by any physical process in your “toy” world, you have broken translational symmetry!

• The brick wall is not an object in this case – it’s some magic applied on top of the coordinate system that functions as a momentum dump for objects that hit it. So, if you were to try to shift your physical system with a translation transformation, only the ball would move. The system is no longer the same after the transformation.

• Without translational symmetry, you don’t have conservation of linear momentum!

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Conclusion

• Symmetry in physics is the preservation of some feature under a transformation• These symmetries help classify various interactions, and

allow us to describe “patterns” of nature.• Symmetries help apply constraints to physical laws, and

guide our discovery of new ones• Thanks to Noether’s theorem, symmetry becomes even

more useful due to its associated conservation laws.

• Symmetry helps bring out the elegance of physics!

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cedric flamant. outline what is symmetry? what is symmetry, to a physicist? physical symmetries uses...

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