We share the hope of Wheeler and Feynman that some of the paradoxesof classical and quantum electrodynamics, in particular the infinite self-energy of a point charge, might be avoidable by working with adjunctfields of this kind.

B = a + ib is an arbitrary 6-component bivector (a and b are relativevectors)

The vector derivative isinvertible so tha

There is another language which has some claim to achieve usefulunifications. The use of "differential forms" became popular with physicists,particularly as a result of its use in the excellent, and deservedly influential,"Big Black Book" by Misner, Thorne, and Wheeler. (33) Differential formsare skew multilinear functions, so that, like multivectors of grade k, theyachieve the aim of coordinate independence. By being scalar-valued,however, differential forms of different grades cannot be combined in theway multivectors can in geometric algebra. Consequently, rotors andspinors cannot be so easily expressed in the language of differential forms

review the subject of monogenicfunctions;

operator precedence convention in which anouter or inner product always takes precedence over a geometric product.Thus a A b is taken before the multiplication by i

"inner product," which is necessary to a great deal ofphysics, has to be grafted into this approach through the use of the dualityoperation, and so the language of differential forms never unifies the innerand outer products in the manner achieved by geometric algebra.

Since ~,~ is a positive-definite scalar in the Pauli algebra, we canwrite¢ = p l / 2 R (3.19)Thus, the Pauli spinor 10) can be seen as a (heavily disguised) instructionto rotate and dilate.

l t h o u g h this is less evident in two dimensions, inthree dimensions it is obvious: the r o t o raR = e x p ( - ia/2) = cos(lat/2) - i~-~ sin(lat/2) (3.13)represents a r o t a t i o n of tal radians a b o u t the axis along the direction of a.

s for at'/spaces, whatever dimension.

"bilinear" transformation of a.

R -= exp( -- al 0-20/2) = cos(0/2) - a l 0"2 sin(0/2)

R--- (3.5)l a + b l , / 2 ( l + b . a

and 0-10-20-3 is thus the unit pseudoscalar for 3-dimen-sional space. In view of its properties, we give it the special symbol i:i-= 0-10-za3

Laser scanning tube lenses can be used in telecentric systems to scan a laser spot across a sample.

Note: Eyepiece designs can be used as the imaging lens to improve performance.

Indeed, (72) is the matrix representation of the GAproduct formula (29),

erates as a rotor in essentially the same way as rotors inclassical mechanics. This suggests that the bilinear dependence of observableson the wave function is not unique to quantum mechanics — it is equally naturalin classical mechanics for geometrical reasons

e have seen, every √−1 has a geometric meaning

be reduced to the form U = a + iβif n is an odd integer, or U = α + ib if n is even.

oducts with a common factor,U1 = ca, U2 = bc. (46)Hence (44) gives usU3 = U2U1 = (bc)(ca) = ba,

U1 = a and U2 = b produces a rotation U3 = ba.

x‖U † = U x‖.

Note that, respectively, the two components commute (anticommute) with i,

unit bivector for the b ∧ a-plane by i.

Incidentally, the term versor was coined in the19th century for an operator that can re-verse a direction.

versor, a

Note how this differs from the field invariantF 2 = (E + iB)2 = E2 − B2 + 2i(E·B),

This enables us to write the outer product defined by (6) inthe forma ∧ b = i a × b

The mostimportant example is the expression of the electromagnetic field F in terms ofan electric vector field E and a magnetic vector field B:F = E + iB

International Conference on Clif-ford Algebras and their Applications in Mathematical Physics”

discussed in the sequel to this paper.

homotopy equivalent to Sn i

metric product

Bivectorschange their sign

∧rB

2 = (−1) d(d−1)2 +s,

I = e1 . . . ed = e1 ∧ . . . ∧ e

IR

polyvector

Clifford algebra structure may be defined in an abstract way, with aClifford product. In this case, the vector space of multivectors is simply a peculiar repre-sentation.

(bilateral) ideal

(In the special case where themetric is zero, Cℓ(V ) = ∧ V .)

One defines the Clifford (or geometrical) product of two vectors asu v ≡ u · v + u ∧ v.

The wedge product of a p−mutivector by itself, M ∧ M , is always 0 when p isodd. This is not true when p is even

form ǫαβγ.

canonical (musical) isomorphism between V and its dual V ∗, which extends to theexterior algebras ∧ V and ∧ V ∗;

an (Hodge) duality (1.2.4) in the exterior algebras

exterior algebra [of multi-vectors] ∧ V of V

⊗

e defines the vector space of all tensors

normalized)

τ (s,r)V ≡s⊗ V ∗r⊗ V

An antisymmetric [con-travariant]tensor of type (0; p) will be called a p-vector , more generally a multivector .An antisymmetric [covariant] tensor of type (p; 0) defines a p-form, more generally amultiform (more simply, a form

that the separation of the spin componentsby the magnetic interaction is counteracted by the effect ofthe Lorentz force on the moving particle.

Themagnetic moment of the orbital motion of the electron in anatom is of the same order, but it can be taken to a classicalvalue by increasing the angular momentum quantum num-ber.

the situation is different for X-ray MO effects. Here the magnetic circular dichroism, whichis equivalent to the Faraday ellipticity, is a commonly used technique

Both z { and } { are thereby expressed as an angle

Various experimental techniques for detecting the Kerr rotation and ellipticity have beendiscussed in the literature (see, e.g., Robinson 1963, Jasperson and Schnatterly 1969, Suits1971, Sato 1981). Schoenes (1992) has given a classification of the available techniquesand a discussion of their respective advantages and disadvantages.

m first-principles energy-bandtheory, and even beyond, that it is feasible to make ab initio predictions of MOKE spectra

effect of Fe provides a minute description of theexperimental MOKE

UNiSn is an antiferromagnetic material

m. Daalderop etal. (1988a) predicted a large Kerr rotation of 5j for UNiSn, but did, unfortunately, notpublish a test of the computational method on simpler systems like the elemental 3i metals.

Another goal of MO research whichbecame intensively pursued in the eighties was to extracted information on the electronicstructure of, in particular, lanthanide and actinide compounds.

Most of the accumulatedexperimental MO data have recently been surveyed by Buschow (1988) and by Schoenes(1992),

measured MO spectraof many materials appeared in the last three decades.

become customary to relateMO phenomena to the materials band structure.

microwave magneto-absorption and Faradayeffect in semiconductors (Dresselhaus et al. 1955, Zwerdling and Lax 1957, Lax et al.1959).

nterest in MO recording, which since then has developed into a leading technologicalapplication of MOKE (

to visualizing surface and subsurface magnetic domains developed.

The only exception was the discovery of the MO Kerr effect in paramagneticmetals in an applied field by Majorana (1944

by Lorentz (1884), based on the idea that left- and right-circularly polarizedlight coupled differently t

MO phenomena hat become an important research topic.Quantum mechanics had not yet emerged, therefore the theoretical understanding of thephenomena was completely lacking

time reversal is preservedand inversion symmetry is absent, a nonequilibrium Kerreffect is allowed by symmetry,

Without loss of generality,

both effects only require inversionsymmetry breaking

nonequilibrium

d that the rectificationinduced by the nonlinear Hall effect

Berrycurvature dipole [5] (BCD)

a nonlinearHall effect

even in the absence of spin-orbitcoupling

nondestructive techniques

his showsthat h is 2π-periodic, and therefore of the form h(t)  g(c(t))

R if k  0,0 if k ≥ 1.Proof. This is a restatement of the Poincaré lemma,

ne can check that these operations arewell-defined

A contractible manifold is simply connected.

k , 1

µ  µM be the volume form of M.

∗dx  nµM .

α0  x · ∗dx‖x‖n .

t dxi ),

φ∗1 (α)  α and φ∗0 (α)  0,

contraction ofM onto x0

On somemanifolds all closed forms (of positive degree) are exact, on others this is true onlyin certain degrees

∫M φ∗ (α0 ) is 2π

0 ∫M×[0,1]φ∗ (dα)

ι0 (x)

g dt dxJ , then ι∗0 (α)  ι∗1 (α)  0

( f (x, 1) − f (x, 0)) dxI  ι∗1 (α) − ι∗0 (α)

It can be regarded as an application of Stokes’ theorem, but weshall give a direct proof.

by taking the piece of α involving dt and integrating it over the unit interval

exists a pointx0 in M

homotopy of loops from φ0 to φ1 in the punctured plane

a piece of string moving through the manifold N

A manifold M is said to be contractible

Suppose that φ0 and φ1 are two maps from amanifold M to a manifold N

φ(x, t)  (1 − t)x + tx/‖x‖.

φ0 (x)  x (identity map) and φ1 (x)  0

he theorem also remains valid if the closed ball is replaced by aclosed cube or a similar shape.

∂M with the induced orientation

so dβ  0.

φ∗ (β) be itspullback to M

(relative to some embedding of M into RN )

orientable manifol

subset A

F(x) · n(x)  (−1)N+1FN (x)

F · ∗dx  (F · n)µM .

n this situation it is best to think of F as the flow vectorfield of a fluid, where the direction of F(x) gives the direction of the flow

α  F · dx

a subordinate partition ofunity consisting of functions

dti  gi (t1, . . . , bi , . . . , tn ) − gi (t1 , . . . , ai , . . . , tn )  0

Suppose M  Hn . Then we can writeα n∑i1gi dt1 dt2 · · ·̂ dti · · · d

64 5. INTEGRATION AND STOKES’ THEOREM5.1. Theorem. Let α be a k-form on U and c : R → U a smooth map. Let p : ¯R → Rbe a reparametrization. Then∫c◦pα ∫c α if p preserves the orientation,− ∫c α if p reverses the orientation

e support of α is certainly compact if the manifold M itself is compact.

contained in ψi (Ui )

i) λi ≥ 0 for all

breaking thedifferential form into small pieces a

φm (x) ≥ cm .

em

It follows that on thehypersurface M the Rn -valued n − 1-form ∗dx is equal to the product of n and ascalar n − 1-form ν: ∗dx  nν.

µM (e1, . . . , eN−1)

)  (−1)N+1FN .

form satisfies µM,x (e1 , . . . , eN−1)  1.

f (v1 , v2 , . . . , vn ; 1) is a negatively oriented frame

volume form µM depends on the embedding of M into RN . I

If this frame is positively,resp. negatively, oriented

This defines an n-form ω on M and we must show that ω  µM

he support of α is defined as the set of all points x in M with theproperty that for every open ball B around x there is a y ∈ B ∩ M such that αy , 0

We define an orientationon M by requiring ψ to be orientation-preserving.

orientation induced

µM depends on the embedding of M into RN . It changes if we dilateor shrink or otherwise deform M

volume forms are seldom exact!

det(Dφ(u) du1 du2 · · · dun

µi  ψ∗i (µ) √det(Dψi (t)T Dψi (t)) dt1 dt2 · · · dtn

Gauss map of M

N+1

[n, e1, e2, . . . , eN−1; 1)  [e1, e2, . . . , eN ; 1].

Proof

Choose ε  ±1 such that (n(x), b1, b2, . . . , bn ; ε) is a positively orientedframe of RN .

structure constant of the Pauli algebra,

ut for mathematical reasons 2 × 2 matrices inphysics need to be unitary

The space of spinors is formally defined as the fundamental representation of the Clifford algebra. (

Vπ = 1.3 volts (EOSPACE Inc,Redmond, WA)

The transmissionaxis (TA) of PL is aligned to the initial linear polarization of the optical beam and to the slow-axis

quation (6) is the fundamental result of this Letter. It

! Imð ~E  ~BÞ ¼ _B  E
_E  B

E is odd under parity whileB is even

We consider a pairof such fields which are interchanged by application ofparity:~EðtÞ ¼  ~E0e
i!t

Yet chiral interac-tions require a time-even pseudoscalar,

does not occur within the point electricdipole approximation, but requires expansion to first orderin ka  10
3, where k is the wave vector of the light and ais the size of the molecule. I

introduce a measure of the local density of chirality of the electromagnetic field.

ow w0¶ in which negative refractioncan be seen for one of the polarizations.

pened up where the permittivity is negative

resonant electric dipoles

Also, the bands at this point havefinite group velocity but infinite phasevelocity

Although referred to as Bleft-handed[ mate-rials, I stress that the sense in which thisterm was used has nothing to do with chi-rality. Therefore I prefer to use the expressionBnegatively refracting[ to avoid confusion.

In addition, there are twolongitudinal modes (not shown), one magneticin character and the other electric.

analogy is less perfect,

magneticplasma

The underlying secret of this medium is that both the di-electric function, ´, and the magnetic permeability, m, hap-pen to be negative.

lthough well-established textbook arguments suggest that static electricsusceptibilityχ(0) must be positive in “all bodies,

The schematic layout (top) and false-coloured microscopeimage (bottom) of a typical device for photocapacitance measurements. Scalebar, 200 µm.

ed as a function of the X-ray power density forthe CsPbBr3 (002) peak.

gle with an increasing X-ray powerdensity and indicating more local lattice distortion with increasing chargecarrier generation.

steps at 20 min intervals, and the illumination power values in the plot haveunits of mW cm −2

18.8 mW cm −2

The dashed lines show fits to the photoconductance decay ateach temperature.

at 10 K and 200 K is 1.0 × 10 6 s and 7.8 × 10 2 s

hows no apparent sign of decay during themeasurement timescale

and the red lines represent the recombination paths, with solidand dashed lines representing large and small probability events, respectively.

Difference between conventional and ferroelectric large polarons.

ferroelectric nanodomains at low temperature,

linear dielectric to paraelectric and relaxor ferroelectric under increasingillumination.

ray diffraction studiesreveal that photocarrier-induced structural polarization is present up toa critical freezing temperature

ultralong photocarrier lifetime beyond 106 s. X

when close to the phase transition temperature

Mach–Zehnder intensity modulator is used to generate psoptical pulses of SLED

We use a 1550 nm laser diode(SFL1550S, Thorlabs) and a super luminescent light emitting diode(SLED; DL-CS5169A, Denslight) for the generation of ps opticalpulses.

in the pulsed Sagnac interfer-ometer, lasers with long coherence lengths perform well

oblique-incidence designenables a study for longitudinal and transverse Kerr effects

all-fiber design simplifies optical alignment;17

s low noise and drift-free measurement

, limited by thephotodetector noise

superconducting states can be broken even with smalloptical powers.

ejects all the reciprocal effects, such as linear birefringence andoptical activity unlike conventional MOKE microscopy

100 nrad/√Hz sensitivityusing only a 10 μW optical power without the magnetic field modu-lation.

ime-reversal symmetry breaking (TRSB) in condensed mat-ter systems generally results in magnetism

1 μrad/√Hz sensitivityat a 3 μW

s the favorable properties of a Sagnac interferometer, such as rejection of all the reciprocal effects, a

onceptual symmetries of resistor, capacitor, inductor, and memristor