Plate trick

Last updated

In mathematics and physics, the plate trick, also known as Dirac's string trick (after Paul Dirac, who introduced and popularized it), [1] [2] the belt trick, or the Balinese cup trick, is any of several demonstrations of the idea that rotating an object with strings attached to it by 360 degrees does not return the system to its original state, while a second rotation of 360 degrees, a total rotation of 720 degrees, does. [3] Mathematically, it is a demonstration of the theorem that SU(2) (which double-covers SO(3)) is simply connected. To say that SU(2) double-covers SO(3) essentially means that the unit quaternions represent the group of rotations twice over. [3] A detailed, intuitive, yet semi-formal articulation can be found in the article on tangloids.

Contents

Demonstrations

Resting a small plate flat on the palm, it is possible to perform two rotations of one's hand while keeping the plate upright. After the first rotation of the hand, the arm will be twisted, but after the second rotation it will end in the original position. To do this, the hand makes one rotation passing over the elbow, twisting the arm, and then another rotation passing under the elbow untwists it. [4] [5]

In mathematical physics, the trick illustrates the quaternionic mathematics behind the spin of spinors. [6] As with the plate trick, these particles' spins return to their original state only after two full rotations, not after one.

The belt trick

Leather belt with frame buckle Belt Gurtel.jpg
Leather belt with frame buckle
Dirac belt trick simulation BeltTrick.gif
Dirac belt trick simulation

The same phenomenon can be demonstrated using a leather belt with an ordinary frame buckle, whose prong serves as a pointer. The end opposite the buckle is clamped so it cannot move. The belt is extended without a twist and the buckle is kept horizontal while being turned clockwise one complete turn (360°), as evidenced by watching the prong. The belt will then appear twisted, and no maneuvering of the buckle that keeps it horizontal and pointed in the same direction can undo the twist. Obviously a 360° turn counterclockwise would undo the twist. The surprise element of the trick is that a second 360° turn in the clockwise direction, while apparently making the belt even more twisted, does allow the belt to be returned to its untwisted state by maneuvering the buckle under the clamped end while always keeping the buckle horizontal and pointed in the same direction. [7]

Mathematically, the belt serves as a record, as one moves along it, of how the buckle was transformed from its original position, with the belt untwisted, to its final rotated position. The clamped end always represents the null rotation. The trick demonstrates that a path in rotation space (SO(3)) that produces a 360 degree rotation is not homotopic to a null rotation, but a path that produces a double rotation (720°) is null-homotopic. [3]

Belt trick has been theoretically constructed in 1-d Classical Heisenberg model as a breather solution. [8]

See also

Related Research Articles

<span class="mw-page-title-main">Fermion</span> Type of subatomic particle

In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Fermions have a half-odd-integer spin and obey the Pauli exclusion principle. These particles include all quarks and leptons and all composite particles made of an odd number of these, such as all baryons and many atoms and nuclei. Fermions differ from bosons, which obey Bose–Einstein statistics.

In theories of quantum gravity, the graviton is the hypothetical quantum of gravity, an elementary particle that mediates the force of gravitational interaction. There is no complete quantum field theory of gravitons due to an outstanding mathematical problem with renormalization in general relativity. In string theory, believed by some to be a consistent theory of quantum gravity, the graviton is a massless state of a fundamental string.

<span class="mw-page-title-main">Spinor</span> Non-tensorial representation of the spin group

In geometry and physics, spinors are elements of a complex number-based vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infinitesimal) rotation, but unlike geometric vectors and tensors, a spinor transforms to its negative when the space rotates through 360°. It takes a rotation of 720° for a spinor to go back to its original state. This property characterizes spinors: spinors can be viewed as the "square roots" of vectors.

<span class="mw-page-title-main">Tangloids</span> Mathematical game

Tangloids is a mathematical game for two players created by Piet Hein to model the calculus of spinors.

Charge, parity, and time reversal symmetry is a fundamental symmetry of physical laws under the simultaneous transformations of charge conjugation (C), parity transformation (P), and time reversal (T). CPT is the only combination of C, P, and T that is observed to be an exact symmetry of nature at the fundamental level. The CPT theorem says that CPT symmetry holds for all physical phenomena, or more precisely, that any Lorentz invariant local quantum field theory with a Hermitian Hamiltonian must have CPT symmetry.

<span class="mw-page-title-main">Magnetic monopole</span> Hypothetical particle with one magnetic pole

In particle physics, a magnetic monopole is a hypothetical elementary particle that is an isolated magnet with only one magnetic pole. A magnetic monopole would have a net north or south "magnetic charge". Modern interest in the concept stems from particle theories, notably the grand unified and superstring theories, which predict their existence. The known elementary particles that have electric charge are electric monopoles.

In theoretical physics, twistor theory was proposed by Roger Penrose in 1967 as a possible path to quantum gravity and has evolved into a widely studied branch of theoretical and mathematical physics. Penrose's idea was that twistor space should be the basic arena for physics from which space-time itself should emerge. It has led to powerful mathematical tools that have applications to differential and integral geometry, nonlinear differential equations and representation theory, and in physics to general relativity, quantum field theory, and the theory of scattering amplitudes. Twistor theory arose in the context of the rapidly expanding mathematical developments in Einstein's theory of general relativity in the late 1950s and in the 1960s and carries a number of influences from that period. In particular, Roger Penrose has credited Ivor Robinson as an important early influence in the development of twistor theory, through his construction of so-called Robinson congruences.

The spin–statistics theorem proves that the observed relationship between the intrinsic spin of a particle and the quantum particle statistics of collections of such particles is a consequence of the mathematics of quantum mechanics. In units of the reduced Planck constant ħ, all particles that move in 3 dimensions have either integer spin and obey Bose-Einstein statistics or half-integer spin and obey Fermi-Dirac statistics.

In physics, an anyon is a type of quasiparticle so far observed only in two-dimensional systems. In three-dimensional systems, only two kinds of elementary particles are seen: fermions and bosons. Anyons have statistical properties intermediate between fermions and bosons. In general, the operation of exchanging two identical particles, although it may cause a global phase shift, cannot affect observables. Anyons are generally classified as abelian or non-abelian. Abelian anyons, detected by two experiments in 2020, play a major role in the fractional quantum Hall effect.

<span class="mw-page-title-main">Anti-twister mechanism</span> Special way of connecting two objects through flexible links

The anti-twister or antitwister mechanism is a method of connecting a flexible link between two objects, one of which is rotating with respect to the other, in a way that prevents the link from becoming twisted. The link could be an electrical cable or a flexible conduit.

<span class="mw-page-title-main">Relativistic wave equations</span> Wave equations respecting special and general relativity

In physics, specifically relativistic quantum mechanics (RQM) and its applications to particle physics, relativistic wave equations predict the behavior of particles at high energies and velocities comparable to the speed of light. In the context of quantum field theory (QFT), the equations determine the dynamics of quantum fields. The solutions to the equations, universally denoted as ψ or Ψ, are referred to as "wave functions" in the context of RQM, and "fields" in the context of QFT. The equations themselves are called "wave equations" or "field equations", because they have the mathematical form of a wave equation or are generated from a Lagrangian density and the field-theoretic Euler–Lagrange equations.

<span class="mw-page-title-main">Majorana fermion</span> Fermion that is its own antiparticle

A Majorana fermion, also referred to as a Majorana particle, is a fermion that is its own antiparticle. They were hypothesised by Ettore Majorana in 1937. The term is sometimes used in opposition to a Dirac fermion, which describes fermions that are not their own antiparticles.

<span class="mw-page-title-main">Orientation entanglement</span> Spinor topology

In mathematics and physics, the notion of orientation entanglement is sometimes used to develop intuition relating to the geometry of spinors or alternatively as a concrete realization of the failure of the special orthogonal groups to be simply connected.

<span class="mw-page-title-main">Feynman checkerboard</span> Fermion path integral approach in 1+1 dimensions

The Feynman checkerboard, or relativistic chessboard model, was Richard Feynman’s sum-over-paths formulation of the kernel for a free spin-½ particle moving in one spatial dimension. It provides a representation of solutions of the Dirac equation in (1+1)-dimensional spacetime as discrete sums.

The Foldy–Wouthuysen transformation was historically significant and was formulated by Leslie Lawrance Foldy and Siegfried Adolf Wouthuysen in 1949 to understand the nonrelativistic limit of the Dirac equation, the equation for spin-1/2 particles. A detailed general discussion of the Foldy–Wouthuysen-type transformations in particle interpretation of relativistic wave equations is in Acharya and Sudarshan (1960). Its utility in high energy physics is now limited due to the primary applications being in the ultra-relativistic domain where the Dirac field is treated as a quantised field.

<span class="mw-page-title-main">Nikodem Popławski</span> Polish physicist

Nikodem Janusz Popławski is a Polish theoretical physicist, most widely noted for the hypothesis that every black hole could be a doorway to another universe and that the universe was formed within a black hole which itself exists in a larger universe. This hypothesis was listed by National Geographic and Science magazines among their top ten discoveries of 2010.

<span class="mw-page-title-main">Riemann–Silberstein vector</span> Complex vector of electromagnetic fields

In mathematical physics, in particular electromagnetism, the Riemann–Silberstein vector or Weber vector named after Bernhard Riemann, Heinrich Martin Weber and Ludwik Silberstein, is a complex vector that combines the electric field E and the magnetic field B.

<span class="mw-page-title-main">Bargmann–Wigner equations</span> Wave equation for arbitrary spin particles

In relativistic quantum mechanics and quantum field theory, the Bargmann–Wigner equations describe free particles with non-zero mass and arbitrary spin j, an integer for bosons or half-integer for fermions. The solutions to the equations are wavefunctions, mathematically in the form of multi-component spinor fields.

Complex spacetime is a mathematical framework that combines the concepts of complex numbers and spacetime in physics. In this framework, the usual real-valued coordinates of spacetime are replaced with complex-valued coordinates. This allows for the inclusion of imaginary components in the description of spacetime, which can have interesting implications in certain areas of physics, such as quantum field theory and string theory.

The term Dirac matter refers to a class of condensed matter systems which can be effectively described by the Dirac equation. Even though the Dirac equation itself was formulated for fermions, the quasi-particles present within Dirac matter can be of any statistics. As a consequence, Dirac matter can be distinguished in fermionic, bosonic or anyonic Dirac matter. Prominent examples of Dirac matter are graphene and other Dirac semimetals, topological insulators, Weyl semimetals, various high-temperature superconductors with -wave pairing and liquid helium-3. The effective theory of such systems is classified by a specific choice of the Dirac mass, the Dirac velocity, the gamma matrices and the space-time curvature. The universal treatment of the class of Dirac matter in terms of an effective theory leads to a common features with respect to the density of states, the heat capacity and impurity scattering.

References

  1. Staley, Mark (2010-01-12). "Understanding Quaternions and the Dirac Belt Trick". arXiv: 1001.1778 [physics.pop-ph].
  2. Schiller, Christoph (2021-01-13). "Testing a conjecture on the origin of the standard model". The European Physical Journal Plus. 136 (1): 79. doi:10.1140/epjp/s13360-020-01046-8. ISSN   2190-5444.
  3. 1 2 3 Staley, Mark (May 2010). "Understanding Quaternions and the Dirac Belt Trick". European Journal of Physics. 31 (3): 467–478. arXiv: 1001.1778 . Bibcode:2010EJPh...31..467S. doi:10.1088/0143-0807/31/3/004. S2CID   118533499.
  4. Leonard Susskind. "Advanced Quantum Mechanics, Lecture 5, time point 51:53".
  5. "Actor Performs the Plate Trick".
  6. Charlie Wood (6 Sep 2018). "The Strange Numbers That Birthed Modern Algebra". Quanta Magazine. Retrieved 9 September 2018.
  7. "Dirac Belt Trick". virtualmathmuseum.org. Retrieved September 9, 2018.
  8. Rahul, O. R.; Murugesh, S. (2019-05-01). "Rogue breather modes: Topological sectors, and the 'belt-trick', in a one-dimensional ferromagnetic spin chain". Chaos, Solitons & Fractals. 122: 262–269. arXiv: 1807.01867 . doi:10.1016/j.chaos.2019.02.012. ISSN   0960-0779. S2CID   104292015.