aa r X i v : . [ phy s i c s . pop - ph ] A p r This is the Accepted Manuscript version of an article accepted for publication in European Journal of Physics. Neither the European PhysicalSociety nor IOP Publishing Ltd is responsible for any errors or omissions in this version of the manuscript or any version derived from it. TheVersion of Record is available online at https://doi.org/10.1088/1361-6404/ab895d.
Chirality Through Classical Physics
Chris L. Lin
Department of Physics, University of Houston, Houston, TX 77204-5005 (Dated: April 20, 2020)Chirality, or handedness, is a topic that is common in biology and chemistry, yet is rarely discussedin physics courses. We provide a way of introducing the topic in classical physics, and demonstratethe merits of its inclusion – such as a simple way to visually introduce the concept of symmetriesin physical law – along with giving some simple proofs using only basic matrix operations, therebyavoiding the full formalism of the three-dimensional point group.
I. INTRODUCTION
Chirality is a topic that spans several sciences, from the helicity of DNA in biology to the existence of organicenantiomers in chemistry [1], to the optical rotation of liquid crystal displays [2, 3] and the handedness of radioactivedecay in physics [4], to name a few. It is a common phenomena of everyday life, from right-handed threaded screwsand twisted ropes to the design choices made in the manufacture of objects catered to a majority right-handedpopulation. As an illustration of the importance of the topic, it has appeared multiple times in the popularBBC-televised annual Christmas lectures given by the Royal Institution of Great Britain [5, 6].A natural starting point to discuss chirality is in optics when discussing the plane mirror, as chirality can broadlybe defined as the study of the effects of replacing an object by its mirror image. The perennial question of a planemirror’s left-right inversion can be used to define chirality and introduce reflection and rotation matrices: this isdone in section II. Having introduced these matrices, in section III we prove some simple facts about reflections androtations using matrix multiplication, and introduce improper rotations. In section IV, we consider the effect ofchirality on interactions, using knots as an example, and make a distinction between the behavior of chiral objectsin physics versus whether the laws of physics are themselves chiral by discussing the violation of parity in the weakinteraction. In section V, we summarize the merits of introducing chirality prior to a course in quantum mechanics,along with conclusions.
II. CHIRALITY DEFINITION
In geometric optics, it is commonly derived that the image is the same height as the object, upright, and at the samedistance from the mirror as the object. Nevertheless it would be a mistake to say that the image is unchanged bythe mirror. The question of the left-right inversion of a plane mirror, which has been discussed extensively in theliterature [7–13], can be modeled with matrices, where R i represents a reflection in the plane perpendicular to axis i ,and R i ( θ ) represents a rotation by angle θ about axis i : R x R y ( π ) = R z − cos π π − sin π π = − (1)which mathematically models the sequence of operations shown in Fig. 1. The frontal inversion z → − z on the RHSis the simplest mathematical way to get an object to face ourselves. However, the most common physical way to getan object to face us is to rotate the object 180 degrees about the y-axis. We have a keen sense of symmetry andnotice that the result after such a rotation differs from the mirror image by a reflection in the x-direction, hence thenomenclature that the mirror inverts left-right [25].FIG. 1: The mirror image of the person is attained directly by the front-back inversion (1 → → → X person = { ( x, y, z ) } T , then due to the bilateral symmetry ofthe person, − xyz = xyz , (2)i.e. the reflection does not change the set of points that comprise the person, and so R x R y ( π ) X person = R z X person becomes just R y ( π ) X person = R z X person , or that the mirror image ( R z X person ) is just a rotation of the object( R y ( π ) X person ), hence the object is achiral. Any object that has a plane of symmetry is achiral, but not all achiralobjects have a plane of symmetry. In the next section we proceed to derive the most general condition for an objectto be achiral: if an object is invariant under a rotation about an axis followed by a reflection in the plane perpen-dicular to the same axis, a combined operation specified by a single axis called an improper rotation, then it is achiral.The quintessential example of a chiral object that often appears in science is the helix. In undergraduate physics,examples include screws, twisted rope, solenoids, kinematics of a charge particles in magnetic fields, circular polar-ization, and the very definition of right-handed coordinate systems. A helix is right-handed if curling the fingers ofyour right-hand around the turns advances you in the direction of your thumb. Alternatively, borrowing from therope and textile sector, a helix is right-handed if when laid vertically and facing you, the turns are moving from thebottom left to the upper right, which is called a ‘Z’ twist due to the shape of the middle of that letter (see Fig. 2).A left-handed helix is called an ‘S’ twist for similar reasons. III. ROTATIONS AND REFLECTIONS
Group theory is generally not part of the standard undergraduate curriculum. However, most students are familiarwith rotation matrices. Such matrices allows one to be precise about the relation between the object and its mirrorimage instead of relying on descriptors such as left, right, turn, and indeed were already used for that purpose in theprevious section, where the slipperiness of words are replaced with the clarity of maths.FIG. 2: Left- and right-handed helices, demonstrated with twisted rope. The left-handed helix on the left cannot berotated into its mirror image, the right-handed helix on the right, hence is chiral.The familiar rotation matrices are given by R x ( α ) = α − sin α α cos α , R y ( α ) = cos α α − sin α α , R z ( α ) = cos α − sin α α cos α
00 0 1 , (3)each of which has determinant equal to one. We construct a general rotation of angle α about an axis in the ( θ, φ )direction, where θ is the polar angle and φ the azimuthal angle, using a similarity transformation: R θ,φ ( α ) = R z ( φ ) R y ( θ ) R z ( α ) R y ( − θ ) R z ( − φ ) , (4)which can be understood as rotating the ( θ, φ ) axis (along with the point that’s rotating about this axis) so that( θ, φ ) aligns with the z -axis, then performing the rotation of α about the z -axis, and then undoing the initial rotation.The determinant of such a matrix is the product of determinants each of which has a value equal to 1, so we haveproven that any rotation has its determinant equal to 1.The reflection matrices are: R x = − , R y = − , R z = − , (5)each of which has determinant equal to negative one compared to their counterparts in (3). The fact that reflectionshave a different determinant than rotations shows that these are in general different operations, although as we haveseen, acting on achiral objects they give the same effect.A general reflection about a plane whose normal is in the ( θ, φ ) direction is given by an argument similar to (4): R θ,φ = R z ( φ ) R y ( θ ) R z R y ( − θ ) R z ( − φ ) . (6)The determinant of such a matrix is the product of determinants which is −
1, so we have proven that any reflectionhas its determinant equal to − P = − − − to reflections: P R θ,φ (180) = R θ,φ . (7)We can now state the most general condition for an object X = xyz to be achiral. If a reflection of the objectis equivalent to a rotation R θ,φ X = R θ ′ ,φ ′ ( α ′ ) X, (8)then clearly R θ,φ X can be superposed on X via the subsequent rotation R − θ ′ ,φ ′ ( α ′ ) = R θ ′ ,φ ′ ( − α ′ ), so that the objectis achiral. However, the common way of defining achirality is if there exists an axis such that rotation about this axis,followed by a reflection in the plane perpendicular to this same axis, leaves the object invariant. This is known asan improper- or roto-rotation. This can again be proven with just matrices. Starting with condition (8) and using R θ,φ = 1: X = R θ,φ R θ ′ ,φ ′ ( α ′ ) X = P R θ,φ (180) R θ ′ ,φ ′ ( α ′ ) X = P R θ ′′ ,φ ′′ ( α ′′ ) X = P R θ ′′ ,φ ′′ (180) R θ ′′ ,φ ′′ (180) R θ ′′ ,φ ′′ ( α ′′ ) X = R θ ′′ ,φ ′′ R θ ′′ ,φ ′′ ( α ′′ + 180) X = R θ ′′ ,φ ′′ R θ ′′ ,φ ′′ ( β ) X. (9)Some particular cases of Eq. (9): if β = 0, then the object has a plane of symmetry [26]. If β = 180, then the objectis symmetric under parity (see Fig. 3a). The reason achirality is defined as an invariance relation on X is becausethe subset of transformations that leave an object X invariant forms a subgroup, the isometry group, and one cancategorize the shape of objects (such as molecules) based on the maximal isometry group to which they belong. Suchconsiderations show that for Eq. (9), β = 360 /n , where n is a positive integer. x z y (a) A molecule that has no planeof symmetry, but can be rotatedinto its mirror image. It isinvariant under parity, and theorigin about which it is invariantis called the inversion center.Formally this is S isometry. (b) A molecule that has noplane of symmetry or inversioncenter. A rotation of 90degrees (about the dottedline), followed by a reflection,leaves the molecule invariant.Formally this is S isometry. FIG. 3: Two examples of achiral molecules.A useful way to think about all this is to view a reflection acting on a chiral object as creating a new object R θ,φ X that lives in a mirror world, where further rotation R θ ′ ,φ ′ R θ,φ X keeps the object in this mirror world as thedeterminant is negative one, i.e., R θ ′ ,φ ′ R θ,φ = R θ ′′ ,φ ′′ , and therefore cannot be attained physically with rotation.Upon another reflection one can exit the mirror world, as a product containing two reflections has determinantpositive one, i.e., R θ ′′ ,φ ′′ R θ ′ ,φ ′ R θ,φ X = R θ ′′′ ,φ ′′′ X . A common demonstration is to show that a left-handed glove,when turned inside-out, becomes a right-handed glove: both a reflection in the z-direction (inside-out) and a reflectionin the x-direction (left-right) move you into the mirror world, where they are related by rotation. Indeed, left-right,top-down, and front-back reflections are all related by rotations.We note that Eq. 9, which takes the form X = ˆ OX , is abstracted to define symmetries in physical law. The set ofall transformations ˆ O that leaves the law of physics X invariant, forms a group. For example, in classical physics,ˆ O represents the Galilean group which includes translations, rotations, and boosts, and X can represent Newton’slaws so that ˆ OX are Newton’s laws seen in the translated, rotated, and boosted frames, respectively. Indeed, modernphysics is often done by selecting a set of symmetries ˆ O that we believe nature respects, and finding laws of physics X that do not change under the symmetry transformations. IV. SYMMETRIES AND INTERACTIONS
Now that we’ve defined chiral objects, we move on to interactions. But before we do, we note that chirality isimportant in chemistry, where for example the left-handed molecule limonene gives an orange its characteristic smellwhile its right-handed version gives a lemon its smell [6]. The fact that we can olfactorily distinguish an orangefrom a lemon means that the detector in our nose, which interacts with the molecule, is itself handed [27]. We havesymmetry of physical law under reflection only if we swap the handedness of both the molecule and detector. Statedanother way, only relative chirality can be detected in an interaction. An analogy in physics is the difference inbinding between a square knot and a granny knot, which we now proceed to discuss.The square or reef knot (Fig. 4a), which is most commonly used to tie shoe-laces, binds two strings (or two ends ofthe same string) together by having a twist of one chirality followed by one of opposite chirality [28]. By contrast, thegranny knot (Fig. 4b) has two twists of the same chirality [29]. Both of these knots look mechanically similar, butit is well-known that the granny knot unravels more easily, although only recently was this explained theoretically indetail [15]. Given that the difference between the two knots are the relative chirality of the top and bottom twists(like-unlike vs like-like), we choose to frame discussion as an interaction between two chiral objects.
T=TL T=TR T=0T=0
A B
T=T load
T=T load (a) A square knot, with a right-handed helix on the bottomand a left-handed helix on top. From left to right, the tensionjumps from 0 to T L as vertex A is crossed, and from T R to 0as vertex B is crossed. (b) The granny knot, with a right-handed helix on the bottomand a right-handed helix on top. Although the top is themirror image of the top of Fig. 4a, the bottom is not, so theentire knot is not the mirror image of Fig. 4a.(c) The mirror image of Fig. 4a. Due to the invariance ofclassical physics under reflection, this knot performs the sameas Fig. 4a. FIG. 4: Although the square and granny knot look similar, they are not mirror reflections of each other, andtherefore can (and do) behave differently.We begin by considering the unravelling of the top helix in Fig. 4a due to T load . Starting from vertex A, friction fromthe left rope (i.e., the rope whose free end is on the left) of tension T L wound around the right rope at vertex B willallow the tension at the loose end of the right rope (i.e. to the right of B) to be zero without this rope sliding to theleft due to the tension T R , thereby unbinding. To calculate the friction needed to prevent slipping we approximatethe right rope at vertex B as a pulley around which the left rope is wound (see Fig. 8), and use the famous capstanformula for the friction of a rope wound around a pulley with a wrap angle of 180 [30]. The friction is T e πµ − T ,which is derived in the appendix. We get T R − ≤ T L e πµ − T L µ ≥ . , (10)where we used the symmetry of the square knot to set T L = T R . This is in good numerical agreement with [16]. Thesame analysis would give the same value of friction for the granny knot. However, this only considers friction fromsliding through the tightened loops and not the twisting of each strand due to friction at the vertices. We will findthat such twisting can be seen as an interaction between the top and bottom helices. To illustrate this, consider Fig.5, which shows the torque exerted at a vertex, with the local twisting of the rope given by the dotted arrow using theright-hand rule. In Fig. 6, the bottom of the left rope is pulled to the left and the resulting velocity of the left rope atvarious points is given by the solid arrow. The resulting torque (dotted arrow) on the right rope will twist this ropelocally in the direction given by the right-hand rule. One can see from Fig. 6 that for the square knot, the net effectis a clockwise twist on the top part of the rope and a counterclockwise twist on the bottom part of the rope. Thistype of strain, where two ends are twisted in opposite directions, is known as torsion [31], and this resists the initialpull velocity. The granny knot is twisted in the same direction at both ends, so it does not cost anything energeticallyfor the rope to start slipping.FIG. 5: A horizontal rope pulled to the right (solid arrow) induces a torque (dotted arrow) on the vertical rope dueto friction from rubbing. The direction of the torque depends on whether the horizontal rope crosses over or underthe vertical rope. (a) Twisting of a square knot. (b) Twisting of a granny knot. FIG. 6: The solid vector denotes the velocity of parts of the left rope when its bottom end is pulled to the left, whilethe dotted vector is the local twist on the right rope caused by friction due to the rubbing of the left rope on theright rope due to the initial pulling motion. The image to the right of each knot is the right rope unwrapped toshow the induced torsion (or lack thereof).That the knots in Fig. 4a and Fig. 4c have the same strength is due to symmetry of classical physics under reflections.We would say that the same laws of physics are obeyed in the mirror world, by which is meant if we were to actuallyconstruct Fig. 4c in the real world and pull it, then the result would be the same as Fig. 4a viewed in a mirror. Moregenerally, if the laws of physics are symmetric under a transformation, then one can perform the transformationfirst and then time evolve, or time evolve and then perform the transformation: time evolution commutes with thesymmetry transformation. This interpretation of symmetry is well-known in both classical and quantum mechanics[17].An example of a symmetry not being obeyed is the parity violation experiment of Wu et al [18]. A spin up Co decays via the weak interaction, ejecting an electron: see Fig. 7. Spin is invariant under parity, but momentumreverses sign, so that the mirror image of Co ’s ejection of an electron with momentum ~p is its ejection of an electronwith momentum − ~p . Therefore a difference in the decay rate between ~p and − ~p in the real world would indicate aviolation of parity [32], and such a difference is indeed found. This stunning result, predicted by Lee and Yang [19],shows that as far as weak interactions are concerned, symmetry arguments such as those used to equate the knots ofFig. 4a and Fig. 4c are invalid and instead each knot must be checked individually, as was done above. Sp S p
FIG. 7: Decay of spin-up colbat-60, where the electron is represented by the small circle, and the antineutrino is notshown. The image on the right is related to the one on the left by parity. However, the right image, if realized in thephysical world, would have a different probability. Therefore party is violated.It was mentioned that chirality can only be detected by a chiral detector [33], and if one were to swap the chiralityof both the object and detector, the same result would occur: in other words, only relative chirality can be detected.However, the violation of parity means one can detect chirality absolutely: left and right can be distinguished in anabsolute sense.
V. CONCLUSIONS
The importance of chirality is reflected in the fact that it receives its own section in introductory organic chemistry[20]. By contrast, in physics, examples of chiral objects are scattered. It is only in quantum mechanics where someformal organization is given to the topic with the introduction of the parity operator. In this paper we argued themerits of having a dedicated section on this topic much earlier than quantum mechanics, to provide a connectionbetween otherwise disparate topics such as the inversion of a plane mirror, solenoid handedness, various right-handrules, circular polarizations, and threaded ropes and fasteners. We have also shown that reflections are a particularlysimple transformation, which makes them ideal for serving as preparation for more abstract concepts such as similaritytransformations, group actions, and symmetry in physical law with its connection to commutativity. Both mirrorsand knots have the additional merit that they visually display chirality. 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Levin, American Journal of Physics , 80 (1991), https://doi.org/10.1119/1.16693, URL https://doi.org/10.1119/1.16693 .[25] For example, in chemistry, one configuration of a molecule may be designated as right-handed, and its mirror image isthen left-handed.[26] It should be noted that any 2D planar object is achiral, where the symmetry plane is the plane of the object.[27] Reflecting a handed object along with an achiral detector produces the opposite-handed object with the same detectorwhich, as we will see, if the laws are symmetric under reflection, will produce the same measurement so that one cannotdistinguish the chirality of the object.[28] To tie this knot a common mnemonic is “right over left, left over right, makes a reef knot both tidy and tight.”[29] To tie this knot “right over left” is followed by another “right over left.”[30] For different methods of approximation and analysis, see [21, 22].[31] Torsion is similar to tension where a bar is subjected to opposite pulling forces at its ends, except the opposite forces arereplaced by opposite torques. A rotational version of Hooke’s law is obeyed that tries to restore the system to its originalangle, which provides additional stability to the square knot.[32] See [4] for a detailed calculation of the Co experiment.[33] An example of a chiral detector is an enzyme whose shape precludes it from binding to a molecule of the opposite-handedness. VI. APPENDIX: THE CAPSTAN EQUATION
To find the tension in a cord wrapped around a rough pulley, equilibrium requires that the sum of the forces in thehorizontal and vertical directions is zero (see Fig. 8). Taylor expanding the tension and keeping terms only to order dθ one gets: T ( θ + dθ ) T ( θ − dθ ) dN ( θ ) µ s dN ( θ ) dθ dθ/ dθ/ FIG. 8: A rope wrapped around a pulley. The rope in the figure represents the left rope in Fig. 4a, and the pulley inthe figure represents the circular cross-section of the right rope in Fig. 4a at vertex B. The size dθ has beenexaggerated for clarity. dTdθ dθ − µ s dN = 0 dN − T dθ = 0 . (11)Solving the second line for dN and substituting into the first line gives dTdθ = µ s TT ( θ ) = T (0) e µ s θ , (12)which is the famous capstan equation [23, 24], which states that a tension of T (0) e µθ at one end can be supportedwith a smaller tension T (0) at the other end, where θ is the wrap angle around the pulley.From Eqn. (11) and Eqn. (12), we now have the distribution of normal force dN ( θ ) = T (0) e µ s θ dθ of the left ropewound around the right rope, so that a maximum bound can be placed on the friction exerted on the right rope: f = Z µ s dN ( θ )= T (0) e πµ − T (0) ..