The deformed paulidirac hamiltonian allows us to study effects of quantum deformation in a class of physical systems, such as a zeemanlike effect, aharonovbohm. The notation chooses to enclose the vector symbol in a surround marker rather than putting an arrow over it. If there is an electron in the magnetic field, then in the. The first obstruction is related to a chiral anomaly. For a free fermion the wavefunction is the product of a plane wave and a dirac spinor, up. Formally, given a vector field v, a vector potential is a vector field a such that. Dirac for quantum physics but we can use it anywhere. A nonsingular potential for the dirac monopole ranjan kumar ghosh bidhannagar college, eb2 salt lake city, calcutta 700064, india palash b. It is consistent with both the principles of quantum mechanics and the theory of special relativity. In the definition of the classical magnetic monopole, the concept of dirac string is used. The dirac string is a considerable embarrassment in monopole theory. On the 2d dirac oscillator in the presence of vector and. Dirac fermions in somraychaudhuri spacetime with scalar and vector potential and the energy momentum distributions parisa sedaghatnia1,a, hassan hassanabadi1,b, faizuddin ahmed2,c 1 faculty of physics. Oct 09, 2016 there are four such matrices labeled by the index \\mu0,1,2,3\ of the vector representation of the lorentz group \o1,3\.
The schrodinger equation is not relativistically invariant. Halfinteger fractional dirac magnetic monopole charges. In conventional vector notation, this is jvj, which is the length of v. So the delta a in the link expresses how the vector potential changes when deforming the dirac string. Ali reza hadjesfandiari university at buffalo school of. The dirac and kleingordon equations with equal scalar and. The dirac string acts as the solenoid in the aharonovbohm effect, and the requirement that the position of the dirac string should not be observable implies the dirac quantization rule. Dirac magnetic monopoles with no singularities researchgate. Dirac notation these notes were produced by david kaplan for phys. This result follows from the fact that the dirac lagrangian is. Theory of magnetic monopoles and electricmagnetic duality. The dirac equation we will try to find a relativistic quantum mechanical description of the electron.
Dirac string is the locus of the points where the vector potential is not wellbehaved. This is analogous to a scalar potential, which is a scalar field whose gradient is a given vector field. Vector potential for diracs string physics stack exchange. Physics 324, fall 2002 dirac notation these notes were produced by david kaplan for phys. In particle physics, the dirac equation is a relativistic wave equation derived by british physicist paul dirac in 1928. Dec 10, 2005 the clue is to show that a deformation of the dirac string is equivalent to a gauge transformation. This dirac string singularity could potentially be detected through the extra phase that the wavefunction of a particle with electric charge q e would acquire if it. It was a scalar function, related to electric eld through e rv. Singlevalued simplyconnected covering groups permitting.
Vector potential of the magnetic monopole physics forums. A coulomb magnetic field which is the curl of a vector potential, i. Dirac and kleingordon equations with equal scalar and vector. After this we rigorously prove that a magnetic monopole field in a static case can only be represented by a scalar potential, not a vector potential as used by dirac. Newest diracstring questions physics stack exchange. The potentials, are singular at the negative axis and the positive axis, respectively. It is assumed that negativeenergymass particles in dirac sea be considered in physical interactions, and that physical space consist of 4dimensional complex space, in another words, each dimension has real and imaginary parts. Br ar 0 always the divergence of a curl of a vector field fr is always zero.
They represent obstructions to the existence of a covariant dirac propagator. Dirac 1931 assumes the concept of vector potential still holds and uses the vector potential in spherical coordinates, r as 1 1 cos tan sin 2 qq mm rr. Jackiw center for theoretical physics massachusetts institute of technology cambridge, ma 0294307 mitctp3327 abstract diracs quantization of magnetic monopole strength is derived without reference to a singular, patched vector potential. Motivated by brane physics, we consider the nonlinear diracborninfeld dbi extension of the abelianhiggs model and study the corresponding cosmic string con. Pdf in most introductory courses on electrodynamics, one is taught. In vector calculus, a vector potential is a vector field whose curl is a given vector field. We analyze the presence of cylindrical symmetric scalar potentials.
The clue is to show that a deformation of the dirac string is equivalent to a gauge transformation. If we adapt this later choice and write the time component of the fourvector potential as g a 0 v t, r, then we end up with two independent potential functions in the dirac and kg equations. Dirac memorial, tallahassee, florida, december 2002. These singularities are known as dirac s string singularities. The easiest way to visualize a magnetic monopole is by considering a pole of. Magnetic monopoles and the quantisation of charge physics. Dirac fermions in somraychaudhuri spacetime with scalar. These are the vector potential v and the scalar potential s. A notation that does this very nicely was invented by the physicist p. Electrodynamics with magnetic monopoles vbn aalborg universitet. I will show that any vector potential for a magnetic monopole has these singularities, and give a physical interpretation to them in section iia. If we adapt this later choice and write the time component of the four vector potential as ga v t r0, g, then we end up with two independent potential functions in the dirac and kg equations. Geometry of the dirac string and its associated vector potential astring. We also demonstrate that the dirac vector potential is actually the vector potential representing the field of a semiinfinite thin solenoid or magnet.
In this section we will describe the dirac equation, whose quantization gives rise to fermionic spin 12particles. It is disconcerting to find that the vector potential that describes a dirac monopole has a string singularity along which the magnetic field is formally infinite, even though we can argue that the string is undetectable. Dirac oscillator in the cosmic string spacetime is considered. On the other hand by introducing the idea of magnetic monopoles, maxwells equations became symmetrical with respect to the magnetic and electric fields, but.
In its free form, or including electromagnetic interactions, it describes all spin1 2 massive particles such as electrons and quarks for which parity is a symmetry. The dirac equation describes the behaviour of spin12 fermions in relativistic quantum. Pdf on the 2d dirac oscillator in the presence of vector. We derive the dirac pauli equation and solve it in the limit of the spin and the pseudospin symmetries. Solution of the deformed dirac equation with vector and. Dirac fermions in somraychaudhuri spacetime with scalar and. Thus, inside the solenoid the vector potential is 2 a 1 n r i. We present arrays relation to pointers and consider the problems arising from their use. In physics, a dirac string is a hypothetical onedimensional curve in space, conceived of by the physicist paul dirac, stretching between two dirac magnetic monopoles with opposite magnetic charges, or from one magnetic monopole out to infinity. The gauge potential cannot be defined on the dirac string, but it is defined everywhere else. Also we would like to have a consistent description of the spin of the electron that in the nonrelativistic theory has to be added by hand. I will show that any vector potential for a magnetic monopole has these singularities, and give a physical interpretation to. This vector potential cannot represent the field of two different physical phenomena at the same time. Dirac and kleingordon equations with equal scalar and.
To do that, we discuss copying in general and consider vectors relation to the lowerlevel notion of arrays. If we adapt this later choice and write the time component of the fourvector potential as ga v t r0, g, then we end up with two independent potential functions in the dirac and kg equations. This is analogous to a scalar potential, which is a scalar field whose gradient is a given vector field formally, given a vector field v, a vector potential is a vector field a such that. The union of the regions in which are welldefined covers the whole of. Since the vector potential is singular along the dirac string, we need two vector potentials to describe electrons in the magnetic field of the magnetic monopole. Finiteenergy dressed stringinspired diraclike monopoles. Dirac notation 1 vectors institute for nuclear theory. The dirac equation with both scalar and vector couplings describing the dynamics of a twodimensional dirac oscillator in the cosmic string spacetime is considered. The dirac string acts as the solenoid in the aharonov. Jackiw center for theoretical physics massachusetts institute of technology cambridge, ma 0294307 mitctp3327 abstract dirac s quantization of magnetic monopole strength is derived without reference to a singular, patched vector potential. Ive been doing some research on magnetic monopoles and i always end up seeing an expression for dirac string vector potential commonly as. Abstract the dirac equation with both scalar and vector couplings describing the dynamics of a twodimensional dirac oscillator in the cosmic string spacetime is considered.
Pal y saha institute of nuclear physics, 1af bidhannagar, calcutta 700064, india august 2002 abstract we propose a new vector potential for the abelian magnetic monopole. These singularities are known as diracs string singularities. Dirac string is the locus of the points where the vector potential is not well behaved. Tomotivatethediracequation,wewillstart by studying the appropriate representation of the lorentz group. L818 letter to the editor we proceed to derive the dirac oscillator equation. Any vector potential a whose curl is equal to b must be singular along some line running from the origin to spatial in. The previous prescription for expressing electric and magnetic fields in terms of the scalar and vector potentials does not uniquely define the potentials. In other words the constant dirac matrices are \o1,3\ invariant.
Recall that a solenoidal field is the curl of some other vector field, e. We derive the diracpauli equation and solve it in the limit of the spin and the pseudospin symmetries. In onedimensional case, the stationary dirac equation describing a particle of mass m in the presence of a vector potential v x and a scalar potential s x reads 6 c. Good advice t his chapter describes how vectors are copied and accessed through subscripting. Indeed, it can be seen that if and, where is an arbitrary scalar field, then the associated electric and magnetic fields are unaffected. Characteristic classes for the index of the dirac family unk a are computed in terms of differential forms on the orbit space of vector potentials under gauge transformations. In the intersection of these regions the vector potentials are related by the gauge transformation, with. Quantization of the free dirac field eduardo fradkin.