- Last 7 days
-
Local file Local file
-
dissipationless spin current
Can this actually make a damping like torque.
-
ny HQ Graphene have led to the advancements such asthe installation of the first commercial automated 2D transfer system inside a glovebox in PI’slab(Fig. 4(c))38, 3
Ralph Lab Glovebox? or at USC?
-
“dry-transfer”
isn't graphene already a dry-transfer?
-
highest SOT efficiency to date17, 20, 21.
Harmonic hall?
-
advantageous reductions in magnetic damping
How compare with Lafo?
-
Born out of a Nobelwinning physics discovery,
Giant magneto resistance? I thought that was hard disks?
-
-
journals.aps.org journals.aps.org
-
Joule heating by the injected current asthe sole source of a thermal gradient and assume ∇T ∝ I 2R s ,where Rs is the sample resistance.
I'd imagine there would be some lag.
-
-
Local file Local fileuntitled1
-
The ISHE converts a spin currentinto an electromotive force ESHE by means of spin–orbit scattering
They assume it's only based on scattering.
-
- Aug 2024
-
en.wikipedia.org en.wikipedia.org
-
Refractive index also varies with wavelength of the light as given by Cauchy's equation: The most general form of Cauchy's equation is n ( λ ) = A + B λ 2 + C λ 4 + ⋯ , {\displaystyle n(\lambda )=A+{\frac {B}{\lambda ^{2}}}+{\frac {C}{\lambda ^{4}}}+\cdots ,} where n is the refractive index, λ is the wavelength
Why only even terms in this equation?
-
-
Local file Local file
-
This isbased on the long-held assumption that a SOT’s effect on the magnetization isindistinguishable from that of a magnetic field.
Wouldn't that only be true for the fieldlike components?
-
- Jul 2024
-
-
ly quasi-TEmodes, but the design process easily extends to quasi-TM modes.
Why quasi?
-
-
opg.optica.org opg.optica.orguntitled8
-
Major dielectricfunctions only can be obtained for materials with cubic, hex-agonal, trigonal, tetragonal, and orthorhombic crystal systems[139]. Such functions can no longer be meaningfully definedfor materials with monoclinic and triclinic crystal systems;
Why?
-
A necessary condition for the underlyingstructure to represent an orthogonal system of electric suscep-tibilities, A must be wavelength independent.
is there something deeper here, or is this just an aesthetic preference. What's wrong with A having wavelength dependence?
-
A rotation ARφ;θ;ψ to diagonalize ε can always be found forsymmetric tensors
For Real symmetric tensors, right?
-
only distin-guishable from χB0 at B ≠ 0
Couldn't you move it over into the B=0 part, so that it venishes by definition?
-
symmetry breaking in space such asby chiral arrangement of matter produce nonsymmetric contri-butions
Then where do you see if it obeys onsager? Shouldn't it only be the time symmetry that produces the non symmetric contributions?
-
which is capable of deter-mining the normalized upper 3 × 3 block of the sample Muellermatrix [22].
What does that correspond to?
-
a truly 3D phenomenon and dispenses withthe requirement of an ideal sheet with infinitesimal small thick-ness.
I wasn't aware this was a requirment.
-
optical Hall effect extends the elec-trical Hall effect to a truly 3D phenomenon
Why can't you do this with electrical Hall?
-
-
Local file Local file
-
ωyωz
suggestive. What is the physical interpretation of these quantities?
-
We are generally not concernedwith the particulars of the resonance,
Why is that?
-
-
Local file Local file
-
C current is now at a much smaller frequency (usually a few kHz), sonow the magnetization is perturbed quasistatically and not driven into resonant
Forbidden? or practical?
-
δ ˆm is very small so its higher-order terms are negligible
Not true if we use a super weak magnet. Which means we may have to do a more complicated treatment.
-
µ0Meff = µ0Ms − 2K⊥/Ms;
Why the factor of two?
-
α d ˆmdt +
missing cross product in this damping term
-
H + µ0Meff( ˆm · ˆZ)ˆZ
Is this making the small angle approximation? It's not really a field right? Is the anisotropy always pointing upwards? Shouldn't it point at some angle phi based on which way the magnet is oriented?
-
-
journals-aps-org.proxy.library.cornell.edu journals-aps-org.proxy.library.cornell.edu
-
τ = τDL + τz
What is \(\tau_z\)? Is this the field like torque?
-
The torques produced by a poly-crystalline thin film must obey Rashba symmetry [18].
Highly confused. What is this?
-
endogenous parameter
What makes these endogenous?
-
esonant linewidth on direct current (dc)
so line width := dc bias STFMR?
-
by extrapolation of the range of the magnetic field
Why extrapolation?
-
Pt/permalloy (Py) a
So, we have these then?
-
ental method used to quantify SOTs inheterostructures that have in-plane magnetic anisotropy[2,5–7].
so STFMR only works for in plane?
-
residual signals
as in statistical "residues"?
-
n at the modulation frequency due to heating.
Why is this a modulation?
-
urrent-induced excitations of a small volume ofmagnetic material with magnetic damping much larger
How do we know that
-
- May 2024
-
Local file Local file
-
onreciprocal phase shift in reflection from anuntwinned single crystal of YBa2Cu307_5, below 50 K (top) and at room temperature(bottom)
Why is it counts?
-
~ 1 mW
that's huge. What's incident on the sample?
-
ultimate limit on our sensitivity will be set by the shotnoise of the laser.
Not of the detector?
-
nti-reflection coated source
why is "antireflection" important here?
-
determine that the optical Kerr effectwas the primary source of error with this laser, producing on offset of ~ 70 \irad,depending on the coupling efficiency in the loop
Why this error?
-
A variant of magneto-optic spectroscopy, X-raycircular dichroism, is an important probe for measuring the orbital contribution tothe overall magnetic moment of materials.
Why xray in particular?
-
he nonlinear magneto-optic Kerr ef-fect(NLMOKE)
what uses this?
-
thecontrast mechanism is well understood and linear in the magnetization, unlike thescanned probe microscopes,
How do we know it's linear?
-
eflection ampUtude|r*pp| may be related to elUpticity, and the phase of r ^ is related to rotation.
shouldn't it be related to neither?
-
lliptically polarized with the planeof polarization in the plane of incidence.
what does that mean?
-
En and Es, must be out of phase
how does that follow?
-
the component paraUel to the surface, ks, is real.
Why ? Don't we have to assume lossles for this?
-
for the Kerr effects, with definitions of the coordinate systemsused for the magnetization and the incident and reflected electric fields.
seems like p and s are swapped?
-
and an analysis basedon the Fresnel equations is no longer valid
Why is that? Can't you separately do each one?
-
i xx along z and i± = ixx ± ie ^
Why are there tildes on everything?
-
nonlocal response to understand a number of opticaleffects, such as optical activity and the anomalous skin effect.
Why is this the case?
-
ctij(u) = since j t = ™ aij(u)Aj
don't see how this follows.
-
<£»•
Shouldn't this be \(\langle \widetilde{\xi'}| m\rangle ^* \)
-
eignekets satisfy |m) =|m), a
by postulate? Or is this proven?
-
no overall rotation
As in rotational momentum?
-
/ f extW = - ^ j ' A ext(t),
does this correctly account for the E field?
-
symmetry of the tensors 6y and /Zy
where did we use this?
-
Magneto-optics only probes an optical penetration depth into the sample, typically ioo A
really? we go through 2nm pt
-
We used the firstconfiguration for most of the experiments because it allowed us to monitor the power
Couldn't you measure at (c) instead?
-
h usually it is in the former position
Why?
-
A = 670 nm [9]. Thisinstrument suffered from a number of difficulties associated with imperfections in thebulk electro-optic modulator (EOM) and with the laser diode source
What were the problems of a bulk EOM?
-
Thermal sources, which have very short coherence lengths, proveto be very effective in this regard
Seriously, that incoherent?
-
Faraday rotation in the fiber loop
even in PM fiber?
-
The 840 nm interferometer has orders ofmagnitude lower amplitude modulation than the 670 nm interferometer,
maybe why our 780 nm interferometer higher noise
-
placing the sample a sixthof the way around the loop, and modulating at three times the proper frequency(3u>m) [9]
why?
-
In the normal-incidence Kerrconfiguration, the sample is capable of backscattering a considerable amount of light,and this effect must be addressed.
Isn't this what you want to measure?
-
if we were to rely solely on this method of polarization controlthe tolerance on the extinction coefficient would be insurmountably high: for phasesensitivity of 1 /zrad, the polarization extinction must be better than 120 dB
Hmm?
-
of other noise sources much earlier than any fundamentalshot noise limit
is the shot noise really a "fundamental limit" because if the light were to encode information more strongly, then signal would be greater for same shot noise, right?
-
21.7 nrad/v/mW Hz
mw should be in numberator, presumabley?
-
if we do not go to thetrouble of specially preparing “sqeezed” photon number states
Hmm, have they tried this?
-
for example, if we choose to interfereorthogonal linear polarization states, for example by rotating one of the fibers inFig. 2.2 by ninety degrees, we will be able to measure the ellipticity epp = ^m (6ptp).This will come at a considerable cost in signal strength, however, because nearly allof the fight will be in the wrong polarization state when it arrives at the polarizer(see Fig. 2.1).
Is there a better way to do this?
-
corresponding to the real andimaginary parts of when we are at normal incidence
is this a \(\theta_k\) vector? what's the hat over it?
-
We set the linear polarization state in one fiber end to be parallel to the optical table,
is this PM fiber?
-
we must maintain a finiteloop length of about 30 m in order to keep the modulation frequency in the MHzrange.
Why not shorter?
-
e Sagnac interferometer to magneto-optic measurements ofmaterials began in 1989
[8] L. Onsager. Reciprocal relations in irreversible processes. II. Phys. Rev., 38:2265- 2279, 1931. [9] B. I. Halperin. The hunt for anyon superconductivity. In Y. Iye and H. Yasuoka, editors, Physics and Chemistry of Oxide Superconductors: Proceedings of the Second ISSP International Symposium. Springer-Verlag, 1992.
-
parasitic effects due to imperfections in the phase modulator will besignificantly reduced near this frequency [10].
Interesting.
-
They showed that proper preparation of the polarization states travelingin the interferometer allowed them to measure magnetic fields via the Faraday effectof the fiber [6, 7
How? Do they use non polarization maintaining fiber?
-
-
Local file Local file
-
conventional superconductors break only gauge symmetry,
what gauge?
-
- Apr 2024
-
en.wikipedia.org en.wikipedia.org
-
rows correspond to irreducible representations
Why is the number of rows not infinite?
Also do these representations have to be faithful? seems like they don't because the trivial representation is a row of the table.
-
-
Local file Local file
-
Z G ◁ G
should be ⊴, right?
-
such, All Irreps of any Abelian Group, e.g., SO(2), are 1-dimensional
How does this follow?
-
1d subspace
1d space not subspace, I presume.
-
{ρg ·⃗v, ∀⃗v ∈ V (i)} = V (i)
shouldn't this be a contained within?
-
Vector Space V ∼= Fd, an
Why is a Irreducible representation tied to a vector space? What about groups not on vector spaces?
-
Aut(H) < Sym(H)
Why isn't an automorphism an element of the symmetric group?
-
x ⋄ y−1
subgroup test seems equivalent to testing the same thing commuted?
-
be a subset of its actions
should this be group elements not actions? Because this assumes a group action?
-
all (continuous) Permutations
What is a continuous permutation?
-
homomorphism that defines how a group acts-on aset, and the set of actions on the Se
Why isn't the second implied by the first?
-
-
geocalc.clas.asu.edu geocalc.clas.asu.edu
-
antisymmetric outer product
I thought this is exterior product, not outer product.
-
- Mar 2024
-
journals.aps.org journals.aps.org
-
to avoid the formation ofvortices in the SC state.
Couldn't you still get a vorticy through the thickness? Maybe just higher energy required because thinner confinement.
-
e ac magnetic field HiSHE generatedvia the iSHE induced charge current JiSHEq is inductively
I presume they really mean ac electric field.
-
suppression of the dampinglike torque generated in the Ptlayer by the inverse spin Hall effect, which can be understood by the changes in spin current transport in thesuperconducting NbN layer.
Could this just be current shunting in the superconductor?
-
-
journals.aps.org journals.aps.org
-
supercurrent is induced along the x axis by enforcing the pairpotential to have a constant phase in a small region close toeach of the two boundaries along x
How does that work?
-
pair potential i = |i| exp(iφi).
cooper pair potential?
-
With respect to thisreference frame, a rotation of the axes by π/2 degrees aboutthe z axis leads to a sign change of ˜αD , while a rotation of π/4degrees changes the tensor to ηso = ˜αD σz.
Not obvious to me
-
m = −γ m × [Heff + Hso ] + αGm × ˙
Seems to be a field like torque that they are considering.
-
textured ferromagnets
What's that? Skyrmions etc?
-
as been observed in systems with broken spatialinversion symmetry and strong spin-orbit coupling (SOC)
Is this the spin orbit torque? What spatial inversion do they mean for a magnet on top of platinum? I assume that inversion is simply the layer order.
-
he maximum achievable spin-orbit torque field is estimated to be on the order of 0.16 mT
This the damping like, or the field-like torque?
-
-
Local file Local file
-
such that the latter induces a Group Action on theVector Space, V , such that the latter induces a Group Action on the Vector Space, V ,
seems duplicate text
-
group G acts-on itself by conjugation
Isn't it the conjugation operator acting on the group?
-
The set of all (continuous) Permutations of the Set, X = Rd, form a Group,Symc(Rd), the Symmetric Group of the set
not obvious to me.
-
G −→ Sym(X)
What is Sym?
-
Group Action on the underlying Vector Space,
Why is this important?
-
Rep of a Group, g ∈ G, on Vector Space
this implies that you could represent a group on something other than a vector space?
-
b) HomF(V, W )G
What do all these decorations mean?
-
L ∈ HomF(V, W )
Why does the homomorphism have to be over a field?
-
-
en.wikipedia.org en.wikipedia.org
-
simple proportion (gain or attenuation), an integral (low-pass filter) and/or derivative (high-pass filter).
That's not obvious to me
-
-
physics.stackexchange.com physics.stackexchange.com
-
signal of strength M2
Where does this come from?
-
-
-
A general structure shouldbe regarded as chiral (Fig. 1e), as long as the total twist is non-zero atany point.
I imagine that it could be oppositely chiral at annother point, and then no longer chiral over all.
-
The structural chirality is not equivalent to thebreaking of inversion
How is it not the same?
-
-
-
bsence of any M, there can be an AFM CD unrelatedto M but proportional to the AFM order L in 𝒫𝒫𝒫𝒫-symmetric AFMs asreported in Cr2O3 (
Why would this happen?
-
ferromagnetic spintronics
notably not antiferrromagnetic spintronics
-
ptical control of chirality andmagnetization,
how to control chirality? I presume they mean during the formation of the molecule.
-
- Feb 2024
-
journals.aps.org journals.aps.org
-
tead a clean signal of a precessing spin-waveexcitation.
That's useful.
-
high field regime, the spin-transfer effectcannot produce a full reversal of the thin-layer moment
Why?
-
which involvedcontinuous ferromagnetic layers
Continuous in what way?
-
pin-polarized electrons flowing from the thinCo layer to the thick layer can switch the moment of thethin layer antiparallel to the thick-layer moment,
How?
-
-
Local file Local file
-
the concept of reciprocity to understand Kerr effect, the useof fluctuation–dissipation theorem, which is another principalresult of linear response theory (that is, the study of irreversiblethermodynamics associated with linear processes that are referredto the equilibrium state of a system) yields an identical result
Does that mean Fluctuation Dissipation carries the same information as Onsager?
-
Reciprocity then requiresthat
not obvious to me.
-
Onsager reciprocity relations and fluctuation–dissipationtheory rely on the assumption that near equilibrium, macroscopicresponse and decay process occur in the same manner as thedecay of equilibrium fluctuations
Question from the past: how could it not?
-
ε ω ε ω′ = ′r r r r( , ; ) ( , ;
Why does this apply to two locations r and r'?
-
V r r( , ; )
potential for what? Photons?
-
ielectric function
Not dielectric constant, notably.
-
spin–orbit coupling and exchange splitting
why?
-
-
Local file Local file
-
he fiber is highly birefringent and the diode light source has8-μm coherence length, only light that couples, after reflectingfrom the sample, between different axes in the fiber willtraverse optical path lengths that differ by less than a coherencelength and interfere coherently at the polarizer
Explains why we need the SLED.
-
tements of the symmetries of the electromagneticfield and its measurement entail that the reflection amplitudesbe considered quantum mechanically
Why??
-
polarization state ± near the source, if not as a plane wave.
Why so complicated? Can't we just use a plane wave source?
-
gen-eral optically active
Things that rotate the polarization of light.
-
the reciprocity theorem
What's that?
-
-
sethna.lassp.cornell.edu sethna.lassp.cornell.edu
-
becomes larger than two
Where two from?
-
- Dec 2023
-
Local file Local file
-
m through long range dipole-dipole couplin
I thought collinear antiferromagnet should have no dipole dipole coupling?
-
readout of magnon density and propagation using photons of visible ligh
How?
-
-
Local file Local file
-
Samples swept with circularly polarized beams
Why does the end ( or maybe it's the start) of the laser path have mixed domains? Is it because they kept the laser there longer, which resulted in thermalized switching?
-
OKE response measures primarily the TM sublattice.
Why? What wavelength are they using?
-
strongertemperature dependence and the T
So, temperature dependence on magnetization gets large as you approach TC?
-
This results in a narrow composition rangewhere the two sublattice
They mean narrow temperature range, not composition range right? Otherwise how does this follow?
-
lting in a net magnetization equal to zero
Looks like you can get complete antiferromagnet.
-
ces. In the case of thelight RE (4f electrons < 7) the two sublattices are exchange-coupledferromagnetically whereas for heavy RE (4f electrons ≥ 7) the twosublattices are exchange-coupled antiferromagnetically, forming aferrimagnet.
Not obvious to me why that's the case.
-
describes deterministic magnetization reversal ofthe material under the beam with no external magnetic field.
Why does one need an antiferrimagnet rather than a normal ferromagnet for this?
-
f GdFeCo (ref. 3), TbCo
Gadolinium and Terbium are the rare earths here.
-
0 fJ is expected to be sufficient
Yes, but requires a large energy source (e.g. 8W laser.) Is this useful for engineers, or just physicists?
-
exhibiting magnetization compensation (andtherefore angular momentum compensation)
compensation?
-
0.4 nm Irinterlayers
Why Iridium?
-
ferromagnetic sample
notably not ferrimagnetic.
-
magnetization switching usingfemto- or picosecond pulsed lasers3
Is this not the all optical thing that they said was good?
-
challenge present theories of AO-HDS
What are these theories, and why are they challenged?
-
all-optical helicity-dependent switching (AO-HDS)
Why does the helicity play a role?
-
manipulating magnetic systems without applied magnetic fields have attracted growing attention
Why is that useful?
-
-
Local file Local file
-
On the other hand,SOT is orthogonal to the magnetization of the free layer, which isexpected to provide “instant-on” switching torque.
No it's not?
-
-
en.wikipedia.org en.wikipedia.org
-
examine the nature of a ∧ b, consider the formula ( a ⋅ b ) 2 − ( a ∧ b ) 2 = a 2 b 2 , {\displaystyle (\mathbf {a} \cdot \mathbf {b} )^{2}-(\mathbf {a} \wedge \mathbf {b} )^{2}=\mathbf {a} ^{2}\mathbf {b} ^{2},}
Where did this come from?
-
-
www.av8n.com www.av8n.com
-
here exist explicit cut-and-dried algorithms for calculating the Hodge dual of B, especially if B is known in terms of components in some basis. See the discussion in reference 13, or see the actual code in reference 14.
it's literally just iB, is it not? Why do I need a reference for that, and why isn't it just directly stated?
-
dot-multiplying by a vector lowers the grade by 1
Sounds like an interior product (such as on wikipedia), rather than an inner product/dot product (as the literature would have it).
-
As another example, in d=3, it converts a vector to a certain “corresponding” pseudoscalar. Meanwhile, in d=4, it converts a vector into the “corresponding” pseudovector (not pseudoscalar)
I think this line should be vector goes to pesudovector, not pseudovector, to be consistant with figure 5.
-
Spinors (aka Pauli spin matrices)
no, a spinnor is the vector of two components that gets acted on by the pauli spin matricies.
-
-
-
The matrix representation is particularly efficient if you have one particular rotation and wish to apply it repeatedly, using it to rotate a large number of vectors. The advantage is most conspicuous in four or more dimensions. See section 7.
So the claim here is that it's only efficient in that case?
-
-
geocalc.clas.asu.edu geocalc.clas.asu.edu
-
the standard Schroedinger wave function is a solution the Schroe-dinger equation (107)
What is the standard Schroedinger wave function? You have to pick some potential, and then there is a schroedinger solution to that potential.
-
definea rotor D by assuming that it satisfies the equationdDdt = 12( emc iB)D
Why do we define this?
-
We can generalize (96) to arbitrary states by interpretingρE = 〈∂t ψ iσ3¯hψ†
not obvious to me how this comes from 96.
-
For a stationary solution with B × s = 0,
Why does a stationary solution have Bxs = 0?
-
we can writeψ = ρ 12 U, where U U † = 1
What is U, and why can we write this? Also, where did the x dependence go?
-
where ψ′ = ψC† is the wave function relative to the alternative quantizationaxis σ′3. The matrix analog of this transformation is a change in matrix repre-sentation for the column spinor Ψ
Why don't we have to multiply it on both sides?
-
i is now the unit pseudoscalar
How do we make this transformation from i' to i?
-
Now write Ψ in the formΨ = ψu,
Why can we do that? Is it that the x, y, and z pauli matricies will rotate things?
-
with complete generality that ψ in (80) is areal even multivector. Now we can reinterpret the σk in ψ as vectors in GAinstead of matrices. Thus, we have established a one-to-one
Why did we have to assume everything was an even multivector?
-
where HS is the Schroedinger hamiltonian
is there anything else in the Schroedinger Hamiltonian? I thought it was only the sigma.B term?
-
a column matrix Ψ
Column vector?
-
σ · a σ · b = a · b I + iσ · (a × b).
order of operations?
-
Indeed, (72) is the matrix representation of the GAproduct formula (29),
not obvious to me.
-
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
Where is the proof?
-
e have seen, every √−1 has a geometric meaning
What meaning?
-
be reduced to the form U = a + iβif n is an odd integer, or U = α + ib if n is even.
Why is this true?
-
oducts with a common factor,U1 = ca, U2 = bc. (46)Hence (44) gives usU3 = U2U1 = (bc)(ca) = ba,
What happened to all the business with left and right multiplication?
-
U1 = a and U2 = b produces a rotation U3 = ba.
This seems no better than the original problem that U3 = U2 U1.
-
x‖U † = U x‖.
How does anti commuting with i imply this?
-
Note that, respectively, the two components commute (anticommute) with i,
Why is that?
-
unit bivector for the b ∧ a-plane by i.
is that a wedge product or geometric product? Maybe it doesn't matter because it's the product of orthogonal vectors.
-
Incidentally, the term versor was coined in the19th century for an operator that can re-verse a direction.
But U seems to do more than just reversing a direction. Seems like it needs a more general term.
-
versor, a
How is this any different from a rotor?
-
Note how this differs from the field invariantF 2 = (E + iB)2 = E2 − B2 + 2i(E·B),
How is this useful? FF* makes more sense.
-
This enables us to write the outer product defined by (6) inthe forma ∧ b = i a × b
How does this follow?
-
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
This was supposed to be an example of adding a complex scalar to a complex vector. However both E and B are vectors. I don't see the relationship.
-
-
Local file Local file1175_1.tif14
-
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.
Why do they hope that?
-
B = a + ib is an arbitrary 6-component bivector (a and b are relativevectors)
Spatial rotation. How does this not also accidentally rotate space into time?
-
The vector derivative isinvertible so tha
is it?
-
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
Is this a sub algebra?
-
review the subject of monogenicfunctions;
what is that?
-
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
hmm
-
"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.
How was it grafted on?
-
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.
Don't see how this follows.
-
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.
not super obvious to me.
-
s for at'/spaces, whatever dimension.
Not obvious from the definition of the geometric product using the wedge.
-
"bilinear" transformation of a.
not the same meaning as bilinear for a two argument function?
-
R -= exp( -- al 0-20/2) = cos(0/2) - a l 0"2 sin(0/2)
This seems to come from outside this text, except that there is no i in the exponential. How do they justify this expression? What is the exponential of a vector?
-
R--- (3.5)l a + b l , / 2 ( l + b . a
Why isn't the rotor simply ab?
-
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
This gives meaning to the hodge with i written in the other geometric algebra treatise by denker.
-
-
www.thorlabs.com www.thorlabs.com
-
Laser scanning tube lenses can be used in telecentric systems to scan a laser spot across a sample.
Why is the laser scan lens before the tube lens?
-
-
www.edmundoptics.com www.edmundoptics.com
-
Note: Eyepiece designs can be used as the imaging lens to improve performance.
In place of?
-