Download Symmetries in Physics: Transformations and Conserved Quantities and more Lecture notes Physics in PDF only on Docsity!
Chapter 5
Basic Principles of Physics
5.1 Symmetry
Symmetry is one of those concepts that occur in our everyday language and also in physics. There is some similarity in the two usages, since, as is usually the case, the physics usage generally grew out of the everyday usage but is more precise. Let’s start with the general usage. Synonyms for symmetry are words like balanced or well formed. We most often use the idea in terms of a work of art. The following 4th century greek statue, Figure 5.1, of a praying boy is a beautiful work of art. This is attributable to the form and balance. The figure has an almost exact bilateral, axial reflection, symmetry. A bilateral symmetry is a well defined mathematical operation on the figure: Establish a mean central axis and place a mirror to reflect every point on the object in the plane plane of the mirror. You recover almost the same figure. In fact a Platonist would attribute the beauty in the piece to the presence of the mathematical symmetry. Of course, for this case, the symmetry is not exact but approximate. These ideas about symmetry can be generalized and at the same time made more specific. In art and in physics, the idea is that you perform some algorithmic or well specified operation to the figure or system of interest. If you recover the same figure or system then you have a symmetry. Later on we will get very specific as to the definition of symmetry but the basic idea that you see here will endure. There is some change that you can make and if after you make the change you have basically the same thing that you started with, you say that you have a symmetry. If you recover almost the same figure or system, you have what is called a slightly broken symmetry or approximate symmetry.
169
170 CHAPTER 5. BASIC PRINCIPLES OF PHYSICS
Figure 5.1: Praying Boy In art, as it will turn out to be the case in physics, there is a sense of beauty associated with balanced or symmetric figures. This ancient greek statue of a praying boy has an approximate bilateral symmetry.
The first issue is to understand the idea of making a change. In order to differentiate the parts of this problem, we will call these changes transfor- mations. There are obviously many transformations that you can perform both in physics and in art. Moving the figure to the side is an especially simple example. The set of operations that are shifting of the figure Is an example of what is called a translation. In art, if the figure is the same after it has been translated, the figure possess translation symmetry; the transformation is a translation and there is a symmetry if the figure is the identical to the original. In most cases in art with translation symmetry, the amount of translation that reproduces the original image is an integer mul- tiple of some fixed amount, see Figure 5.3. This is an example of a discrete translation symmetry. Our earlier example of bilateral transformations or mirror images is also an example of a discrete family of transformations. This is an especially simple family since, if you do the transformation twice, you have not done anything. There are thus only two transformations in the bilateral set: mirror image or leave alone. The case of Figure 5.3, there are many translations that produce a symmetry. In fact, there is an infinite
172 CHAPTER 5. BASIC PRINCIPLES OF PHYSICS
teger multiple of 26 π , reproduce the original image. Again, like the bilateral transformation, after so many of these rotations you can get back to doing nothing. This is a more interesting example of the discrete transformations with a finite number of elements than the bilateral case besides doing noth- ing there are five non-trivial rotations, π 3 , 23 π , π, 43 π , and 53 π. In addition, the snowflake also has a bilateral. In fact since it has the discrete rotations, it actually has several bilateral transformations. These are along axis at 0, 2 π 12 ,^
4 π 12 ,^
6 π 12 ,^
8 π 12 , and^
10 π 12.^ These being combinations of the bilaterals and rotations.
Figure 5.4: Snowflakes Snowflakes provide an excellent natural example of a system with a discrete rotation symmetry. It also has a bilateral symmetry and, since it has a rotational symmetry, actually has several bilaterals.
As stated earlier, symmetry is a change to a system or, in the case of art, a figure or a statue that is not an important change. From these examples it is important to realize that to have a symmetry, you need a set of changes to the figure and then a criteria for these not being an important change. In the case of art, the criteria for not being important is that the pieces fall on top of each other. You could have a much more relaxed definition of unimportant change. For example consider the world of three sided figures whose sides are straight lines, triangles. If your criteria for unimportant change is that after the transformation you still have a triangle, then any transformation short of opening or bending one of the sides will be a symmetry. You could have a more restrictive criteria such as that the triangles be similar. In this case, rotations and rescaling all lengths would be a symmetry but changing
5.1. SYMMETRY 173
the size of one of the sides and not the others would not. It is important to keep in mind that the concept of symmetry is a two step process – a family of transformations and a rule about what is an important change. Although we did not discuss it in those terms, we have already had an example of a symmetry in physics when we looked at the change in scale when we discussed dimensional analysis, see Section 2.5. If we change the scale of length, all the numbers change but the things that happen still happen; it doesn’t matter whether you make the measurements in the cgs system, the mks system, or english system, the physics is the same. We can use this as a rather loose definition of what we mean by a symmetry in physics. As we develop our vocabulary more fully, we can make this definition much more precise. For all of the discussion so far we have defined the transformations as changes to the figure; rotate the figure by 26 π. With the example of change in scale, we can see a different but clearly equivalent approach. Instead of stretching the figure, we can just use a smaller length scale to discuss its size. In the old perspective, you can also look at it as if all lengths increased and the unit of length stayed the same. Here you now say that the figure stays the same and the unit of length changes. This is the difference between the active and the passive view of a transformation. In the active view, you change the figure, in the passive view the figure is left unchanged but the observers perspective is changed. In the active view, you then have another perspective
Figure 5.5: Spiral The spiral is generated by stretching the radius as you rotate. This is an example of a situation in which you combine two simple transformations to generate a figure with symmetry.
in symmetry. You can use the transformation to generate a figure that will automatically be symmetric. An extreme example of a symmetry is the infinitely long straight line. It satisfies bilateral symmetry about every
5.2. THE NATURE OF SYMMETRY IN PHYSICS 175
5.2.1 Discrete Transformations
These are changes that can only be applied in discrete steps. Bilateral or mirror symmetry about a plane is an example from art. For the snow flakes, the rotations at θ = n π 3 for n = 1, 2 .... is an example of a family of discrete transformations that produce a symmetry. What do you think happens for n = 0? Is this the same as n = 6? The rule is that, once you have a set of transformations, the set must contain all combinations of the transformations for the set to be complete. The example in physics that corresponds to bilateral symmetry is called a spatial inversion which is to replace places in one directions by their op- posite. In a world with on space dimension, replace x by −x. In a world with three spatial directions, replace (x, y, z) with (−x, y, z). This is like placing a mirror in the plane y = 0, z = 0. This is obviously a discrete transformation. You also note that, if it is applied twice, there is no change. It is said to be a discrete transformation of cycle two; it has two elements, do nothing, the identity transformation, and the inversion. There are many discrete transformations of cycle two: if you have identical particles, you can interchange the particles, you can invert the time, you can do a spatial inversion along the y or z axis, ... There are, of course, discrete transformations with cycles higher than two. The snowflake example from art carries over to physics. Rotations about the origin by an angle of (^2) nπ is an example of a discrete transformation with n cycles. You can also have a family of discrete transformations that have an infinite number of elements. In one spatial dimension, you can shift the origin by a fixed amount, a. You can do this any number of times generating a set of transformations that has a countable infinite number of members. It is important to realize that the method by which the mem- bers of a family of discrete transformations are labeled must itself be a discrete set of labels and that the members of a discrete set of transformations cannot be labeled by a continuous variable.
5.2.2 Continuous Transformations
Continuous transformations are changes that can be applied for arbitrarily small changes. The labeling of the transformations is a continuous param- eter. Rotations about a point are a valuable example. In art, a world of concentric rings would enjoy a symmetry for rotations about the center point. These changes in angle can take any value from zero to 2π. This idea
176 CHAPTER 5. BASIC PRINCIPLES OF PHYSICS
is carried over to physics. In a three dimensional space, rotations about an axis are a family of transformations. These transformations are an example of continuous transformations. Other obvious examples are translations in space and time. Changes in the scale of length discussed in Sections 1.5.1, and 2.5.2 is also a continuous set of transformations. Again it is impor- tant to realize that a continuous family of transformations can only be labeled by a continuous variable. It is possible to make a discrete family of transformations from sub- sets of continuous transformations such as the set of rotations used in the snowflake example of Figure 5.4 in Section 5.1. Of course, the reverse pro- cess is not possible; you cannot make a continuous family of transformations from a subset of a discrete family no matter how large the set of discrete transformations.
5.2.3 Identity Transformation
The identity transformation is the one that leaves everything alone. The example n = 0 in the discrete case above is an identity transformation. Note that n = 6m where m = 1, 2 , 3 ... is also the identity and we already had it in the set of transformations. In fact, any transformation in which n > 6 is the same as the transformation n′^ = mod 6 (n).
5.2.4 Examples of symmetry in situations like physics
You are planning a trip between Austin and College Station. There are several routes.
Figure 5.7: Paths to Texas A&M Miles to AM
178 CHAPTER 5. BASIC PRINCIPLES OF PHYSICS
Figure 5.8: Action trajectory Trajectory 2
Space Translation:
Shift the origin of the coordinate system.
x → x′^ = x + a (5.2)
Time Translation:
Shift the start of the time.
t → t′^ = t + a (5.3)
To be a symmetry we will require that the physics before and after the shift is the same. I have not carefully defined what I mean by ”the same.” I will do so shortly.
Newton’s Action at a Distance Law of Gravitation
The law of force that describes the gravitational influence of one body, say body 2, on another body, say body 1, is
F^ ~ 1 , 2 = G m^1 m^2 |~r 2 − ~r 1 |^3
× (~r 2 − ~r 1 ) (5.4)
Similarly, the gravitational force of body 1 on body 2 can be found by interchanging the labels of particles 1 and 2.
5.3. EXAMPLES OF SYMMETRY IN PHYSICS 179
Figure 5.9: Space Reflection Space Reflection
Figure 5.10: Space Translation Space Translation
F^ ~ 2 , 1 = G m^2 m^1 |~r 1 − ~r 2 |^3
× (~r 1 − ~r 2 ) (5.5)
Thus if you are operating at the level of the forces you have that if you interchange particles 1 and 2, i. e. change the labels 1 and 2, 1 ↔ 2 and get F~ 1 , 2 → − F~ 2 , 1 This is a discrete transformation. If for some reason you are interested in the forces, this is not a symmetry. It is actually a manifestation of the Law of Action Reaction. In other words, we construct the Law of Gravitation so that it obeys the Law of Action Reaction. On the other hand, if you look at the entire set of equations without the forces, there is no change.
5.4. SYMMETRY AND ACTION 181
x′^ = x + a t′^ = t (5.8)
This implies that v′^ = v. Thus
S′(x′ f , t′ f , x′ 0 , t′ 0 ; path′) =
(x ∑′ f ,t′ f )
path′,(x′ 0 ,t′ 0 )
(m
v′^2 2 )∆t
(x ∑f ,tf )
path,(x 0 ,t 0 )
(m v^2 2
)∆t
= S(xf , tf , x 0 , t 0 ; path) (5.9)
If action is the basis of all physics, then we have a natural definition of a symmetry of a physical system. A physical system has a symmetry if there is a way to modify the system and yet there is no significant change in the action. It is important to be careful about the meaning of significant in this sentence. For most purposes the value of the action is not important. The action primary role is to select a path from the infinity of possibilities. In this sense, we can as a first step assert that the system is symmetric if the system before and after the change still selects the same path as the natural path. You again have to be careful because the same path is actually the same path as seen in the modified system. An example might help clarify this.
Harmonic Oscillator and Symmetry
The harmonic oscillator is one of the most important physical systems. We will discuss the physics of this system in greater detail in a later section, Section 6.2, but for now will use it as another example in which to examine the role of symmetry in a physical system. For now just think of of it as a physical system that goes back and forth. The Lagrangian for the harmonic oscillator is
L(v, x) = KE − P E = m
v^2 2 − k
x^2 2
where k is the spring constant and m is the mass and both are given con-
stants and have the dimension k dim = (^) TimeMass 2 and, of course, m is a mass. Note
182 CHAPTER 5. BASIC PRINCIPLES OF PHYSICS
that, if these are the only two dimensional constants that are available, then you cannot make a length but you can make a time. If you rescale the distances by an amount λ, as follows:
x → x′^ = λx t → t′^ = t (5.11)
which implies that
v → v′^ = ∆x′ ∆t′^
= λ ∆x ∆t
= λv (5.12)
The Lagrangian for the new system is
L′(v′, x′) = KE′^ −P E′^ = m v′^2 2
−k x′^2 2
= mλ^2 ( v^2 2
−k x^2 2
) = λ^2 L(v, x) (5.13)
So that
S P ath′ ′ (x′ 0 , t′ 0 ; x′ f , t′ f ) =
(x ∑′ f ,t′ f )
path′,(x′ 0 ,t′ 0 )
(m
v′^2 2 − k
x′^2 2 )∆t′
= λ^2
(x ∑f ,tf )
path,(x 0 ,t 0 )
(m v^2 2
− k x^2 2
)∆t
= λ^2 SP ath(x 0 , t 0 ; xf , tf ) (5.14)
where Path’ is the Path that is at the rescaled distances
x′(t′) = λx(t) (5.15)
Figure 5.12: Rescale Oscillator Rescale Oscillator
184 CHAPTER 5. BASIC PRINCIPLES OF PHYSICS
= S(xf , tf , x 0 , t 0 ; path) − ma
x ∑f ,tf
path,x 0 ,t 0
v∆t +
m a^2 2
) x ∑f^ ,tf
path,x 0 ,t 0
∆t
= S(xf , tf , x 0 , t 0 ; path) − ma(xf − x 0 ) +
m a^2 2
(tf − t 0 ) (5.18)
The last two terms are independent of path. Therefore the path selection process the selects the least path in S will select the transformed path in S′. The action changes under the transformation but in an unimportant way. This is not a symmetry and there is no associated conserved quantity.When we implement this for special relativity it will become a symmetry.
5.4.3 More on Symmetry and Action
The easiest way to guarantee that the action is symmetric under a set of transformations is to construct it only from the form invariants for that set of transformations. In fact, it is a necessary and sufficient condition that the action is symmetric that it be composed of only form invariants for that set of transformations. As an example consider the action for a satellite of mass m in orbit around the earth. Locating the earth at the origin, the action is
S(~x 0 , t 0 , ~xf , tf ; path) =
~x ∑f ,tf
P ath,~x 0 ,t 0
(m ~v^2 2
)∆t (5.19)
This action is composed of ~v^2 which is a form invariant for rotations about the origin. r is the distance from the origin and it is also a form invariant for rotations. Obviously ∆t is a form invariant for rotations. Thus this action has a symmetry that is the set of transformations that are the rotations about the origin.
5.4.4 Noether’s Theorem
For every continuous transformation that is connected to the identity that is a symmetry, no important change, there is a conserved quantity. Noether’s Theorem also tells you how to construct the conserved quantity. When I tell you what the question is and thus when a change is important, I can tell you how to construct the conserved quantity.
5.4. SYMMETRY AND ACTION 185
Space translation Symmetry
The conserved quantity that is associated with situations with space trans- lation symmetry is called linear momentum. In certain cases it is ~p = m~v but not all the time. I will tell you when those cases are.
Rotation translation symmetry
The conserved quantity that is associated with situations with space rotation symmetry is called angular momentum. Rotations are a vector quantity. Again in certain cases it is L~ = m~r × ~v.
Time translation Symmetry
The conserved quantity that is associated with situations with time trans- lation symmetry is called energy. This is actually the case all the time but the form of the energy may change.
Galilean Invariance
This is almost a symmetry classically and becomes a full blown symmetry in the modern language. First, let’s discuss what the transformation is. There is no experiment that can be performed that can measure the velocity of an moving observer. We can detect the presence of accelerations and measure the relative velocity between two bodies but we cannot measure absolute velocities. Another way to say the same thing is that, if you are not accelerating, you are always at rest in your own rest frame. In the language of transformations, all the laws of physics must be in- variant under a transformation of the form
~x → ~x′^ = ~x + R~ + ~vt t → t′^ = t (5.20)
where R~ and ~v are constants that are the parameters that label the contin- uous transformations. They can be interpreted in terms of two coordinate systems this can be interpreted as the difference in the measurements of two relatively displaced and relatively moving coordinate systems. Although this is a continuous symmetry that is connected with the iden- tity, it is not a symmetry classically. I will explain this later. Since this is
5.4. SYMMETRY AND ACTION 187
but also that the terms from the end points vanish. This part simply selects the natural path. To understand the end points consider an example, the simple translation. In this case δx is simply a number that is added to all points in the path.
δx(tf ) = δx(t 0 ) = a (5.23)
or
δL δv
|xnat(t)
δx
tf
δL δv
|xnat(t)
δx
t 0
δL δv
|xnat(t)
tf
δL δv
|xnat(t)
tf
a
(5.24) Setting this to zero, yields ( δL δv
|xnat(t)
tf
δL δv
|xnat(t)
tf
But δLδv |xnat(t) is what you would define as the momentum. It is the mo- mentum when you use the usual Lagrangian. Thus this is nothing more than the statement that momentum is conserved.
p(tf ) = p(t 0 ) (5.26)
This is a special case of a general theorem called Noether’s Theorem. Given any transformation that can be connected with the identity transformation, no change, by a continuous parameter. There will always be a conserved quantity. In the above example the transformation is translation. In the limit a → 0 you have no translation and thus no change and the identity transformation. In this case, the conserved quantity is the linear momentum. Another way of looking at this result is that, once you have selected the natural path and if you include the end point variations, the action is a function of the end points only. If the symmetry transformation changes the end points you have
δSN at(x 0 , t 0 ; xf , tf ) =
δSN at δx 0 δx 0 +
δSN at δxf δxf +
δSN at δt 0 δt 0 +
δSN at δtf δtf (5.27)
In the case of translations,
δx(tf ) = δx(t 0 ) = a (5.28)
188 CHAPTER 5. BASIC PRINCIPLES OF PHYSICS
and all the δti are zero. Thus we get δS δxf
δS δx 0 = p = constant (5.29)
An Example
For the free particle,
Snatural = m
(xf − x 0 )^2 2(tf − t 0 )
p =
δS δxf = m
(xf − x 0 ) (tf − t 0 ) = mv (5.31)
since v is a constant. We noted above that the satellite in orbit is a case that is invariant under rotations about the origin. This set of transformations is a continuous set and thus there is a conserved quantity. In this case we call it the angular momentum. The construction of this conserved quantity involves cumber- some notation because it only makes sense in a system with at least two spatial dimensions and thus involves vector notation. In addition, it is com- putationally difficult to find an expression for the natural path. But note that the free particle Lagrangian is also composed only of form invariants for rotations about the origin. Thus this set of transformations is also a sym- metry for this case. The analysis is still cumbersome because of the vector notation. I am aware that you will not be able to reproduce this analysis. All that I ask is that you follow it. We will work in two spatial dimensions. For this case the action is
S(~x 0 , ~t 0 ; ~xf , ~tf ) =
~x ∑f ,~tf
NaturalPath,~x 0 ,~t 0
m ~v^2 2 ∆t (5.32)
and as we see is composed of only form invariants not only of translations in space and time but also for rotations. The quantity ~v^2 is invariant under rotations. For the natural path the action is
Snatural = m
(~xf − ~x 0 )^2 2(tf − t 0 )
