Superfluid He-3. Thought of the Day 130.0

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At higher temperatures 3He is a gas, while below temperature of 3K – due to van der Walls forces – 3He is a normal liquid with all symmetries which a condensed matter system can have: translation, gauge symmetry U(1) and two SO(3) symmetries for the spin (SOS(3)) and orbital (SOL(3)) rotations. At temperatures below 100 mK, 3He behaves as a strongly interacting Fermi liquid. Its physical properties are well described by Landau’s theory. Quasi-particles of the 3He (i.e. 3He atoms “dressed” into mutual interactions) have spin equal to 1/2 and similar to the electrons, they can create Cooper pairs as well. However, different from electrons in a metal, 3He is a liquid without a lattice and the electron-phonon interaction, responsible for superconductivity, can not be applied here. As the 3He quasiparticles have spin, the magnetic interaction between spins rises up when the temperature falls down until, at a certain temperature, Cooper pairs are created – the coupled pairs of 3He quasiparticles – and the normal 3He liquid becomes a superfluid. The Cooper pairs produce a superfluid component and the rest, unpaired 3He quasiparticles, generate a normal component (N -phase).

A physical picture of the superfluid 3He is more complicated than for superconducting electrons. First, the 3He quasiparticles are bare atoms and creating the Cooper pair they have to rotate around its common center of mass, generating an orbital angular momentum of the pair (L = 1). Secondly, the spin of the Cooper pair is equal to one (S = 1), thus superfluid 3He has magnetic properties. Thirdly, the orbital and spin angular momenta of the pair are coupled via a dipole-dipole interaction.

It is evident that the phase transition of 3He into the superfluid state is accompanied by spontaneously broken symmetry: orbital, spin and gauge: SOL(3)× SOS(3) × U(1), except the translational symmetry, as the superfluid 3He is still a liquid. Finally, an energy gap ∆ appears in the energy spectrum separating the Cooper pairs (ground state) from unpaired quasiparticles – Fermi excitations.

In superfluid 3He the density of Fermi excitations decreases upon further cooling. For temperatures below around 0.25Tc (where Tc is the superfluid transition temperature), the density of the Fermi excitations is so low that the excitations can be regarded as a non-interacting gas because almost all of them are paired and occupy the ground state. Therefore, at these very low temperatures, the superfluid phases of helium-3 represent well defined models of the quantum vacua, which allows us to study any influences of various external forces on the ground state and excitations from this state as well.

The ground state of superfluid 3He is formed by the Cooper pairs having both spin (S = 1) and orbital momentum (L = 1). As a consequence of this spin-triplet, orbital p-wave pairing, the order parameter (or wave function) is far more complicated than that of conventional superconductors and superfluid 4He. The order parameter of the superfluid 3He joins two spaces: the orbital (or k space) and spin and can be expressed as:

Ψ(k) = Ψ↑↑(kˆ)|↑↑⟩ + Ψ↓↓(kˆ)|↓↓⟩ + √2Ψ↑↓(kˆ)(|↑↓⟩ + |↓↑⟩) —– (1)

where kˆ is a unit vector in k space defining a position on the Fermi surface, Ψ↑↑(kˆ), Ψ↓↓(kˆ) a Ψ↑↓(kˆ) are amplitudes of the spin sub-states operators determined by its projection |↑↑⟩, |↓↓⟩ a (|↑↓⟩ + |↓↑⟩) on a quantization axis z.

The order parameter is more often written in a vector representation as a vector d(k) in spin space. For any orientation of the k on the Fermi surface, d(k) is in the direction for which the Cooper pairs have zero spin projection. Moreover, the amplitude of the superfluid condensate at the same point is defined by |d(k)|2 = 1/2tr(ΨΨH). The vector form of the order parameter d(k) for its components can be written as:

dν(k) = ∑μ Aνμkμ —– (2)

where ν (1,2,3) are orthogonal directions in spin space and μ (x,y,z) are those for orbital space. The matrix components Aνμ are complex and theoretically each of them represents possible superfluid phase of 3He. Experimentally, however, only three are stable.

phasediagramLooking at the phase diagram of 3He we can see the presence of two main superfluid phases: A – phase and B – phase. While B – phase consists of all three spin components, the A – phase does not have the component (|↑↓⟩ + |↓↑⟩). There is also a narrow region of the A1 superfluid phase which exists only at higher pressures and temperatures and in nonzero magnetic field. The A1 – phase has only one spin component |↑↑⟩. The phase transition from N – phase to the A or B – phase is a second order transition, while the phase transition between the superfluid A and B phases is of first order.

The B – phase occupies a low field region and it is stable down to the lowest temperatures. In zero field, the B – phase is a pure manifestation of p-wave superfluidity. Having equal numbers of all possible spin and angular momentum projections, the energy gap separating ground state from excitation is isotropic in k space.

The A – phase is preferable at higher pressures and temperatures in zero field. In limit T → 0K, the A – phase can exist at higher magnetic fields (above 340 mT) at zero pressure and this critical field needed for creation of the A – phase rises up as the pressure increases. In this phase, all Cooper pairs have orbital momenta orientated in a common direction defined by the vector lˆ, that is the direction in which the energy gap is reduced to zero. It results in a remarkable difference between these superfluid phases. The B – phase has an isotropic gap, while the A – phase energy spectrum consists of two Fermi points i.e. points with zero energy gap. The difference in the gap structure leads to the different thermodynamic properties of quasiparticle excitations in the limit T → 0K. The density of excitation in the B – phase falls down exponentially with temperature as exp(−∆/kBT), where kB is the Boltzman constant. At the lowest temperatures their density is so low that the excitations can be regarded as a non-interacting gas with a mean free path of the order of kilometers. On the other hand, in A – phase the Fermi points (or nodes) are far more populated with quasiparticle excitations. The nodes orientation in the lˆ direction make the A – phase excitations almost perfectly one-dimensional. The presence of the nodes in the energy spectrum leads to a T3 temperature dependence of the density of excitations and entropy. As a result, as T → 0K, the specific heat of the A – phase is far greater than that of the B – phase. In this limit, the A – phase represents a model system for a vacuum of the Standard model and B – phase is a model system for a Dirac vacuum.

In experiments with superfluid 3He phases, application of different external forces can excite the collective modes of the order parameter representing so called Bose excitations, while the Fermi excitations are responsible for the energy dissipation. Coexistence and mutual interactions of these excitations in the limit T → 0K (in limit of low energies), can be described by quantum field theory, where Bose and Fermi excitations represent Bose and Fermi quantum fields. Thus, 3He has a much broader impact by offering the possibility of experimentally investigating quantum field/cosmological theories via their analogies with the superfluid phases of 3He.

Fermi Surface Singularities

In ideal Fermi gases, the Fermi surface at p = pF = √2μm is the boundary in p-space between the occupied states (np = 1) at p2/2m < μ and empty states (np = 0) at p2/2m > μ. At this boundary (the surface in 3D momentum space) the energy is zero. What happens when the interaction between particles is introduced? Due to interaction the distribution function np of particles in the ground state is no longer exactly 1 or 0. However, it appears that the Fermi surface survives as the singularity in np. Such stability of the Fermi surface comes from a topological property of the one-particle Green’s function at imaginary frequency:

G-1 = iω – p2/2m + μ —– (1)

Let us for simplicity skip one spatial dimension pz so that the Fermi surface becomes the line in 2D momentum space (px,py); this does not change the co-dimension of zeroes which remains 1 = 3−2 = 2−1. The Green’s function has singularities lying on a closed line ω = 0, p2x + p2y = p2F in the 3D momentum-frequency space (ω,px,py). This is the line of the quantized vortex in the momentum space, since the phase Φ of the Green’s function G = |G|e changes by 2πN1 around the path embracing any element of this vortex line. In the considered case the phase winding number is N1 = 1. If we add the third momentum dimension pz the vortex line becomes the surface in the 4D momentum-frequency space (ω,px,py,pz) – the Fermi surface – but again the phase changes by 2π along any closed loop empracing the element of the 2D surface in the 4D momentum-frequency space.

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Fermi surface is a topological object in momentum space – a vortex loop. When the chemical potential μ decreases the loop shrinks and disappears at μ < 0. The point μ = T = 0 marks the Lifshitz transition between the gapless ground state at μ > 0 to the fully gapped vacuum at μ < 0.

The winding number cannot change by continuous deformation of the Green’s function: the momentum-space vortex is robust toward any perturbation. Thus the singularity of the Green’s function on the Fermi surface is preserved, even when interaction between fermions is introduced. The invariant is the same for any space dimension, since the co-dimension remains 1.

The Green function is generally a matrix with spin indices. In addition, it may have the band indices (in the case of electrons in the periodic potential of crystals). In such a case the phase of the Green’s function becomes meaning-less; however, the topological property of the Green’s function remains robust. The general analysis demonstrates that topologically stable Fermi surfaces are described by the group Z of integers. The winding number N1 is expressed analytically in terms of the Green’s function:

N= tr ∮C dl/2πi G(μ,p) ∂lG-1(μ,p) —– (2)

Here the integral is taken over an arbitrary contour C around the momentum- space vortex, and tr is the trace over the spin, band and/or other indices.

The Fermi surface cannot be destroyed by small perturbations, since it is protected by topology and thus is robust to perturbations. But the Fermi surface can be removed by large perturbations in the processes which reproduces the processes occurring for the real-space counterpart of the Fermi surface – the loop of quantized vortex in superfluids and superconductors. The vortex ring can continuously shrink to a point and then disappear, or continuously expand and leave the momentum space. The first scenario occurs when one continuously changes the chemical potential from the positive to the negative value: at μ < 0 there is no vortex loop in momentum space and the ground state (vacuum) is fully gapped. The point μ = 0 marks the quantum phase transition – the Lifshitz transition – at which the topology of the energy spectrum changes. At this transition the symmetry of the ground state does not changes. The second scenario of the quantum phase transition to the fully gapped states occurs when the inverse mass 1/m in (1) crosses zero.

Similar Lifshitz transitions from the fully gapped state to the state with the Fermi surface may occur in superfluids and superconductors. This happens, for example, when the superfluid velocity crosses the Landau critical velocity. The symmetry of the order parameter does not change across such a quantum phase transition. In the non-superconduting states, the transition from the gapless to gapped state is the metal-insulator transition.

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The Lifshitz transitions involving the vortex lines in p-space may occur be- tween the gapless states. They are accompanied by the change of the topology of the Fermi surface itself. The simplest example of such a phase transition discussed in terms of the vortex lines is provided by the reconnection of the vortex lines.

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Grand Unification Theory/(Anti-GUT): Emerging Symmetry, Topology in a Momentum Space. Thought of the Day 129.0

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Quantum phase transition between two ground states with the same symmetry but of different universality class – gapless at q < qc and fully gapped at q > qc – as isolated point (a) as the termination point of first order transition (b)

There are two schemes for the classification of states in condensed matter physics and relativistic quantum fields: classification by symmetry (GUT scheme) and by momentum space topology (anti-GUT scheme).

For the first classification method, a given state of the system is characterized by a symmetry group H which is a subgroup of the symmetry group G of the relevant physical laws. The thermodynamic phase transition between equilibrium states is usually marked by a change of the symmetry group H. This classification reflects the phenomenon of spontaneously broken symmetry. In relativistic quantum fields the chain of successive phase transitions, in which the large symmetry group existing at high energy is reduced at low energy, is in the basis of the Grand Unification models (GUT). In condensed matter the spontaneous symmetry breaking is a typical phenomenon, and the thermodynamic states are also classified in terms of the subgroup H of the relevant group G. The groups G and H are also responsible for topological defects, which are determined by the nontrivial elements of the homotopy groups πn(G/H).

The second classification method reflects the opposite tendency – the anti Grand Unification (anti-GUT) – when instead of the symmetry breaking the symmetry gradually emerges at low energy. This method deals with the ground states of the system at zero temperature (T = 0), i.e., it is the classification of quantum vacua. The universality classes of quantum vacua are determined by momentum-space topology, which is also responsible for the type of the effective theory, emergent physical laws and symmetries at low energy. Contrary to the GUT scheme, where the symmetry of the vacuum state is primary giving rise to topology, in the anti-GUT scheme the topology in the momentum space is primary while the vacuum symmetry is the emergent phenomenon in the low energy corner.

At the moment, we live in the ultra-cold Universe. All the characteristic temperatures in our Universe are extremely small compared to the Planck energy scale EP. That is why all the massive fermions, whose natural mass must be of order EP, are frozen out due to extremely small factor exp(−EP/T). There is no matter in our Universe unless there are massless fermions, whose masslessness is protected with extremely high accuracy. It is the topology in the momentum space, which provides such protection.

For systems living in 3D space, there are four basic universality classes of fermionic vacua provided by topology in momentum space:

(i)  Vacua with fully-gapped fermionic excitations, such as semiconductors and conventional superconductors.

(ii)  Vacua with fermionic excitations characterized by Fermi points – points in 3D momentum space at which the energy of fermionic quasiparticle vanishes. Examples are provided by the quantum vacuum of Standard Model above the electroweak transition, where all elementary particles are Weyl fermions with Fermi points in the spectrum. This universality class manifests the phenomenon of emergent relativistic quantum fields at low energy: close to the Fermi points the fermionic quasiparticles behave as massless Weyl fermions, while the collective modes of the vacuum interact with these fermions as gauge and gravitational fields.

(iii)  Vacua with fermionic excitations characterized by lines in 3D momentum space or points in 2D momentum space. We call them Fermi lines, though in general it is better to characterize zeroes by co-dimension, which is the dimension of p-space minus the dimension of the manifold of zeros. Lines in 3D momentum space and points in 2D momentum space have co-dimension 2: since 3−1 = 2−0 = 2. The Fermi lines are topologically stable only if some special symmetry is obeyed.

(iv) Vacua with fermionic excitations characterized by Fermi surfaces. This universality class also manifests the phenomenon of emergent physics, though non-relativistic: at low temperature all the metals behave in a similar way, and this behavior is determined by the Landau theory of Fermi liquid – the effective theory based on the existence of Fermi surface. Fermi surface has co-dimension 1: in 3D system it is the surface (co-dimension = 3 − 2 = 1), in 2D system it is the line (co- dimension = 2 − 1 = 1), and in 1D system it is the point (co-dimension = 1 − 0 = 1; in one dimensional system the Landau Fermi-liquid theory does not work, but the Fermi surface survives).

The possibility of the Fermi band class (v), where the energy vanishes in the finite region of the 3D momentum space and thus zeroes have co-dimension 0, and such topologically stable flat band may exist in the spectrum of fermion zero modes, i.e. for fermions localized in the core of the topological objects. The phase transitions which follow from this classification scheme are quantum phase transitions which occur at T = 0. It may happen that by changing some parameter q of the system we transfer the vacuum state from one universality class to another, or to the vacuum of the same universality class but different topological quantum number, without changing its symmetry group H. The point qc, where this zero-temperature transition occurs, marks the quantum phase transition. For T ≠ 0, the second order phase transition is absent, as the two states belong to the same symmetry class H, but the first order phase transition is not excluded. Hence, there is an isolated singular point (qc, 0) in the (q, T) plane, or the end point of the first order transition. The quantum phase transitions which occur in classes (iv) and (i) or be- tween these classes are well known. In the class (iv) the corresponding quantum phase transition is known as Lifshitz transition, at which the Fermi surface changes its topology or emerges from the fully gapped state of class (i). The transition between the fully gapped states characterized by different topological charges occurs in 2D systems exhibiting the quantum Hall and spin-Hall effect: this is the plateau-plateau transition between the states with different values of the Hall (or spin-Hall) conductance. The less known transitions involve nodes of co-dimension 3 and nodes of co-dimension 2.

The Fallacy of Deviant Analyticity. Thought of the Day 128.0

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Carnap’s thesis of pluralism in mathematics is quite radical. We are told that “any postulates and any rules of inference [may] be chosen arbitrarily”; for example, the question of whether the Principle of Selection (that is, the Axiom of Choice (AC)) should be admitted is “purely one of expedience” (Logical Syntax of Language); more generally,

The [logico-mathematical sentences] are, from the point of view of material interpretation, expedients for the purpose of operating with the [descriptive sentences]. Thus, in laying down [a logico-mathematical sentence] as a primitive sentence, only usefulness for this purpose is to be taken into consideration.

So the pluralism is quite broad – it extends to AC and even to ∏01-sentences. There are problems in maintaining ∏01-pluralism. One cannot, on pain of inconsistency, think that statements about consistency are not “mere matters of expedience” without thinking that ∏01-statements generally are not mere “matters of expedience”. The question of whether a given ∏01-sentence holds is not a mere matter of expedience; rather, such questions fall within the provenance of theoretical reason. One reason is that in adopting a ∏01-sentence one could always be struck by a counter-example. Other reasons have to do with the clarity of our conception of the natural numbers and with our experience to date with that structure. On this basis, for no sentence of first-order arithmetic is the question of whether it holds a mere matter of experience. Certainly this is the default view from which one must be moved.

What does Carnap have to say that will sway us from the default view, and lead us to embrace his radical form of pluralism? In approaching this question it is important to bear in mind that there are two general interpretations of Carnap. According to the first interpretation – the substantive – Carnap is really trying to argue for the pluralist conception. According to the second interpretation – the non-substantive – he is merely trying to persuade us of it, that is, to show that of all the options it is most “expedient”.

The most obvious approach to securing pluralism is to appeal to the work on analyticity and content. For if mathematical truths are without content and, moreover, this claim can be maintained with respect to an arbitrary mathematical system, then one could argue that even apparently incompatible systems have null content and hence are really compatible (since there is no contentual-conflict).

Now, in order for this to secure radical pluralism, Carnap would have to first secure his claim that mathematical truths are without content. But, he has not done so. Instead, he has merely provided us with a piece of technical machinery that can be used to articulate any one of a number of views concerning mathematical content and he has adjusted the parameters so as to articulate his particular view. So he has not secured the thesis of radical pluralism. Thus, on the substantive interpretation, Carnap has failed to achieve his end.

This leaves us with the non-substantive interpretation. There are a number of problems that arise for this version of Carnap. To begin with, Carnap’s technical machinery is not even suitable for articulating his thesis of radical pluralism since (using either the definition of analyticity for Language I or Language II) there is no metalanguage in which one can say that two apparently incompatible systems S1 and S2 both have null content and hence are really contentually compatible. To fix ideas, consider a paradigm case of an apparent conflict that we should like to dissolve by saying that there is no contentual-conflict: Let S1 = PA + φ and S2 = PA + ¬φ, where φ is any arithmetical sentence, and let the metatheory be MA = ZFC. The trouble is that on the approach to Language I, although in MT (metatheory) we can prove that each system is ω-complete (which is a start since we wish to say that each system has null content), we can also prove that one has null content while the other has total content (that is, in ω-logic, every sentence of arithmetic is a consequence). So, we cannot, within MT articulate the idea that there is no contentual-conflict. The approach to Language II involves a complementary problem. To see this note that while a strong logic like ω-logic is something that one can apply to a formal system, a truth definition is something that applies to a language (in our modern sense). Thus, on this approach, in MT the definition of analyticity given for S1 and S2 is the same (since the two systems are couched in the same language). So, although in MT we can say that S1 and S2 do not have a contentual-conflict this is only because we have given a deviant definition of analyticity, one that is blind to the fact that in a very straightforward sense φ is analytic in S1 while ¬φ is analytic in S2.

Now, although Carnap’s machinery is not adequate to articulate the thesis of radical pluralism in a given metatheory, under certain circumstances he can attempt to articulate the thesis by changing the metatheory. For example, let S1 = PA + Con(ZF + AD) and S2 = PA + ¬Con(ZF + AD) and suppose we wish to articulate both the idea that the two systems have null content and the idea that Con(ZF + AD) is analytic in S1 while ¬Con(ZF + AD) is analytic in S2. No single metatheory (on either of Carnap’s approaches) can do this. But it turns out that because of the kind of assessment sensitivity, there are two metatheories MT1 and MT2 such that in MT1 we can say both that S1 has null content and that Con(ZF + AD) is analytic in S1, while in MT2 we can say both that S2 has null content and that ¬Con(ZF + AD) is analytic in S2. But, of course, this is simply because (any such metatheory) MT1 proves Con(ZF+AD) and (any such metatheory) MT2 proves ¬Con(ZF+AD). So we have done no more than reflect the difference between the systems in the metatheories. Thus, although Carnap does not have a way of articulating his radical pluralism (in a given metalanguage), he certainly has a way of manifesting it (by making corresponding changes in his metatheories).

As a final retreat Carnap might say that he is not trying to persuade us of a thesis that (concerning a collection of systems) can be articulated in a given framework but rather is trying to persuade us to adopt a thorough radical pluralism as a “way of life”. He has certainly shown us how we can make the requisite adjustments in our metatheory so as to consistently manifest radical pluralism. But does this amount to more than an algorithm for begging the question? Has Carnap shown us that there is no question to beg? He has not said anything persuasive in favour of embracing a thorough radical pluralism as the “most expedient” of the options. The trouble with Carnap’s entire approach is that the question of pluralism has been detached from actual developments in mathematics.

Super-Poincaré Algebra: Is There a Case for Nontrivial Geometry in the Bosonic Sector?

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In one dimension there is no Lorentz group and therefore all bosonic and fermionic fields have no space-time indices. The simplest free action for one bosonic field φ and one fermionic field ψ reads

S = γ∫dt [φ.2 – i/2ψψ.] —– (1)

We treat the scalar field as dimensionless and assign dimension cm−1/2 to fermions. Therefore, all our actions will contain the parameter γ with the dimension [γ] = cm. (1) provides the first example of a supersymmetric invariant action and is invariant w.r.t. the following transformations:

δφ = −iεψ, δψ = −εφ ̇ —– (2)

The infinitesimal parameter ε anticommutes with fermionic fields and with itself. What is really important about transformations (2) is their commutator

δ2δ1φ = δ2(−ε1ψ) = iε1ε2φ ̇

δ1δ2φ = iε2ε1φ ̇ ⇒ [δ2, δ1] φ = 2iε1ε2φ ̇ —– (3)

Thus, from (3) we may see the main property of supersymmetry transformations: they commute on translations. In our simplest one-dimensional framework this is the time translation. This property has the followin form in terms of the supersymmetry generator Q:

{Q, Q} = −2P —– (4)

The anticommutator (4), together with

[Q, P] = 0 —– (5)

describe N = 1 super-Poincaré algebra in d = 1. The structure of N-extended super-Poincaré algebra includes N real super-charges QA , A = 1, . . . , N with the following anti commutators:

{QA, QB} = −2δABP, [QA, P] = 0 —– (6)

Let us stress that the reality of the supercharges is very important, as well as having the same sign in the r.h.s. of QA, QB ∀ QA.

From (2) we see that the minimal N = 1 supermultiplet includes one bosonic and one fermionic field. A natural question arises: how many components do we need, in order to realize the N-extended superalgebra (6)? In order to mimic the transformations (2) for all N supertranslations

δφi = −iεA(LA)i ψ, δψ = −εA (RA)iφ ̇i —– (7)

Here the indices i = 1,…,db and iˆ = 1,…,df count the numbers of bosonic and fermionic components, while (LA)i and (RA)i are N arbitrary, for the time being, matrices. The additional conditions one should impose on the transformations (7) are

  • they should form the N-extended superalgebra (6)
  • they should leave invariant the free action constructed from the involved fields.

When N > 8 the minimal dimension of the supermultiplets rapidly increases and the analysis of the corresponding theories becomes very complicated. For many reasons, the most interesting case seems to be the N = 8 supersymmetric mechanics. Being the highest N case of minimal N-extended supersymmetric mechanics admitting realization on N bosons (physical and auxiliary) and N fermions, the systems with eight supercharges are the highest N ones, among the extended supersymmetric systems, which still possess a nontrivial geometry in the bosonic sector. When the number of supercharges exceeds 8, the target spaces are restricted to be symmetric spaces.

Super Lie Algebra

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A super Lie algebra L is an object in the category of super vector spaces together with a morphism [ , ] : L ⊗ L → L, often called the super bracket, or simply, the bracket, which satisfies the following conditions

Anti-symmetry,

[ , ] + [ , ] ○ cL,L = 0

which is the same as

[x, y] + (-1)|x||y|[y, x] = 0 for x, y ∈ L homogenous.

Jacobi identity,

[, [ , ]] + [, [ , ]] ○ σ + [, [ , ]] ○ σ2 = 0,

where σ ∈ S3 is a three-cycle, i.e. taking the first entity of [, [ , ]] to the second, and the second to the third, and then the third to the first. So, for x, y, z ∈ L homogenous, this reads

[x + [y, z]] + (-1)|x||y| + |x||z|[y, [z, x]] + (-1)|y||z| + |x||z|[z, [x, y]] = 0

It is important to note that in the super category, these conditions are modifications of the properties of the bracket in a Lie algebra, designed to accommodate the odd variables. We can immediately extend this definition to the case where L is an A-module for A a commutative superalgebra, thus defining a Lie superalgebra in the category of A-modules. In fact, we can make any associative superalgebra A into a Lie superalgebra by taking the bracket to be

[a, b] = ab – (-1)|a||b|ba,

i.e., we take the bracket to be the difference τ – τ ○ cA,A, where τ is the multiplication morphism on A.

A left A-module is a super vector space M with a morphism A ⊗ M → M, a ⊗ m ↦ am, of super vector spaces obeying the usual identities; that is, ∀ a, b ∈ A and x, y ∈ M, we have

a (x + y) = ax + ay

(a + b)x = ax + bx

(ab)x  = a(bx)

1x = x

A right A-module is defined similarly. Note that if A is commutative, a left A-module is also a right A-module if we define (the sign rule)

m . a = (-1)|m||a|a . m

for m ∈ M and a ∈ A. Morphisms of A-modules are defined in the obvious manner: super vector space morphisms φ: M → N such that φ(am) = aφ(m) ∀ a ∈ A and m ∈ M. So, we have the category of A-modules. For A commutative, the category of A-modules admits tensor products: for M1, M2 A-modules, M1 ⊗ M2 is taken as the tensor product of M1 as a right module with M2 as a left module.

Turning our attention to free A-modules, we have the notion of super vector kp|q over k, and so we define Ap|q := A ⊗ kp|q where

(Ap|q)0 = A0 ⊗ (kp|q)0 ⊕ A1 ⊗ (kp|q)1

(Ap|q)1 = A1 ⊗ (kp|q)0 ⊕ A0 ⊗ (kp|q)1

We say that an A-module M is free if it is isomorphic (in the category of A-modules) to Ap|q for some (p, q). This is equivalent to saying that M contains p even elements {e1, …, ep} and q odd elements {ε1, …, εq} such that

M0 = spanA0{e1, …, ep} ⊕ spanA11, …, εq}

M1 = spanA1{e1, …, ep} ⊕ spanA01, …, εq}

We shall also say M as the free module generated over A by the even elements e1, …, eand the odd elements ε1, …, εq.

Let T: Ap|q → Ar|s be a morphism of free A-modules and then write ep+1, …., ep+q for the odd basis elements ε1, …, εq. Then T is defined on the basis elements {e1, …, ep+q} by

T(ej) = ∑i=1p+q eitij

Hence T can be represented as a matrix of size (r + s) x (p + q)

T = (T1 T2 T3 T4)

where T1 is an r x p matrix consisting of even elements of A, T2 is an r x q matrix of odd elements, T3 is an s x p matrix of odd elements, and T4 is an s x q matrix of even elements. When we say that T is a morphism of super A-modules, it means that it must preserve parity, and therefore the parity of the blocks, T1 & T4, which are even and T2 & T3, which are odd, is determined. When we define T on the basis elements, the basis elements precedes the coordinates tij. This is important to keep the signs in order and comes naturally from composing morphisms. In other words if the module is written as a right module with T acting from the left, composition becomes matrix product in the usual manner:

(S . T)(ej) = S(∑i eitij) = ∑i,keksiktij

hence for any x ∈ Ap|q , we can express x as the column vector x = ∑eixi and so T(x) is given by the matrix product T x.

Fallibilist a priori. Thought of the Day 127.0

Figure-1-Peirce's-ten-classes-of-sign-represented-in-terms-of-three-core-functions-and

Kant’s ‘transcendental subject’ is pragmatized in this notion in Peirce, transcending any delimitation of reason to the human mind: the ‘anybody’ is operational and refers to anything which is able to undertake reasoning’s formal procedures. In the same way, Kant’s synthetic a priori notion is pragmatized in Peirce’s account:

Kant declares that the question of his great work is ‘How are synthetical judgments a priori possible?’ By a priori he means universal; by synthetical, experiential (i.e., relating to experience, not necessarily derived wholly from experience). The true question for him should have been, ‘How are universal propositions relating to experience to be justified?’ But let me not be understood to speak with anything less than profound and almost unparalleled admiration for that wonderful achievement, that indispensable stepping-stone of philosophy. (The Essential Peirce Selected Philosophical Writings)

Synthetic a priori is interpreted as experiential and universal, or, to put it another way, observational and general – thus Peirce’s rationalism in demanding rational relations is connected to his scholastic realism posing the existence of real universals.

But we do not make a diagram simply to represent the relation of killer to killed, though it would not be impossible to represent this relation in a Graph-Instance; and the reason why we do not is that there is little or nothing in that relation that is rationally comprehensible. It is known as a fact, and that is all. I believe I may venture to affirm that an intelligible relation, that is, a relation of thought, is created only by the act of representing it. I do not mean to say that if we should some day find out the metaphysical nature of the relation of killing, that intelligible relation would thereby be created. [ ] No, for the intelligible relation has been signified, though not read by man, since the first killing was done, if not long before. (The New Elements of Mathematics)

Peirce’s pragmatizing Kant enables him to escape the threatening subjectivism: rational relations are inherent in the universe and are not our inventions, but we must know (some of) them in order to think. The relation of killer to killed, is not, however, given our present knowledge, one of those rational relations, even if we might later become able to produce a rational diagram of aspects of it. Yet, such a relation is, as Peirce says, a mere fact. On the other hand, rational relations are – even if inherent in the universe – not only facts. Their extension is rather that of mathematics as such, which can be seen from the fact that the rational relations are what make necessary reasoning possible – at the same time as Peirce subscribes to his father’s mathematics definition: Mathematics is the science that draws necessary conclusions – with Peirce’s addendum that these conclusions are always hypothetical. This conforms to Kant’s idea that the result of synthetic a priori judgments comprised mathematics as well as the sciences built on applied mathematics. Thus, in constructing diagrams, we have all the possible relations in mathematics (which is inexhaustible, following Gödel’s 1931 incompleteness theorem) at our disposal. Moreover, the idea that we might later learn about the rational relations involved in killing entails a historical, fallibilist rendering of the a priori notion. Unlike the case in Kant, the a priori is thus removed from a privileged connection to the knowing subject and its transcendental faculties. Thus, Peirce rather anticipates a fallibilist notion of the a priori.