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BLED WORKSHOPS
IN PHYSICS
VOL. 20, NO. 2
Proceedings to the 22nd Workshop
What Comes Beyond . . . (p. 223)
Bled, Slovenia, July 6–14, 2019
13 Second Quantization as Cross Product
N.S. Mankoč Borštnik1 and H.B.F. Nielsen2 ?
1Department of Physics, University of Ljubljana,
SI-1000 Ljubljana, Slovenia
2Niels Bohr Institute, University of Copenhagen, Blegdamsvej 15–21,
Copenhagen Ø, Denmark
Abstract. In the contributions [4,5] of this proceedings the new way of the second quan-
tization of fermions is proposed, inspired by the fact that the Clifford and Grassmann
algebra by themselves offer basis in internal space, presented as creation operators on the
corresponding vacuum state, which together with their Hermitian conjugated annihilation
partners fulfill all the requirements for the second quantized fermions, provided that the
part of the basis in the ordinary space is orthogonal. In the Hilbert space of indefinite
number of fermions it is assumed that each fermion has to distinguish from all the others
either in ordinary or in internal space or in both spaces. The purpose of this contribution is
to generalize this last requirement for either fermions or bosons.
Povzetek. V prispevkih [4,5] tega zbornika predstavita avtorja nov način druge kvanti-
zacije fermionov. Cliffordov in Grassmanov prostor ponudita namreč bazo v notranjem
prostoru fermionov, ki jo določajo kreacijski operatorji na vakuumskem stanju, ti pa sku-
paj s Hermitsko andjungiranimi operatorji (annihilacijskimi operatorji) izpolnjujejo vse
Diracove zahteve za fermione v drugi kvantizaciji pod pogojem, da je baza v prostoru
gibalnih količin ortogonalna. V Hilbertovem prostoru nedoločenega števila fermionov mora
vsakemu fermionu ustrezati drugačen notranji prostor ali drugačna gibalna količina, V
drugi kvantizaciji je Hilbertov prostor direkten produkt neskončne množice Hilbertovih
prostorov za izbrano vrednost gibalne količine. Namen tega prispevka je posplošiti ta drugi
del zahteve tako za fermione kot za bozone.
Keywords: second quantization, bosons, fermions, cross product
13.1 Introduction
We present in this contribution the possibility to make a new step in the new way
of the second quantization of fermions, presented in the contributions [4,5] of this
proceedings, for indefinite number of fermions and bosons.
It is the purpose of the present discussion to seek to use such a formulation
of second quantized theories to generalize them to possibly quite new types of
second quantization like theories. This is inspired from the type of theory put
forward by one of us as being unification theory of spin, charges and families [1–5]
? H.B. Nielsen presented the talk.
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224 N.S. Mankoč Borštnik and H.B.F. Nielsen
Second quantization as Cross product
It is rather trivial and welknown that a second quantized (free) theory of
bosons has a second quantized Hilbert space, that can be written as a Cartesian
cross product over an (infinite) set of (smaller) Hilbert spaces, each of which is
attached for example to the momentum, and tells how many particles have just
this momentum.
Simplest case: A scalar without internal degrees of freedom
If we think of a charged scalar - like π+ - it may be natural to even include
in our “momentum” also the sign of the energy and use that as the ‘factorisation
parameter” p. We like to do it as abstract and general as possible, so we now use
the letter p and you can think of it as “(factorization) parameter” or as momentum
as you like.
In the π+ case we take the “factorization parameter” p to be:
p = (~p, sign(E)). (13.1)
The general form as factorized space:
The Hilbert space for the second quantized boson system can always be
written like
H =
⊗
p
Hp. (13.2)
In the π+ example, where p = (~p, sign(E)), the Hilbert space Hp is actually
that of harmonic oscillator for which the number operator counts the number of
π+ particles with just the p-specification p.
(Here we stepped too fast over the Dirac sea for bosons problem, but that is not
so crucial just now; just think of antiparticles instead, when formally sign(E) < 0.)
Dream of generalization(s)
In the formulation as the Cartesian product
H =
⊗
p
Hp (13.3)
one could dream about making a new - and perhaps interesting theory - by
replacing the Hilbert spaces that are factors in the Cartesian product such asHp
by some Hilbert spaces with a different structure, e.g. different dimensionality.
E.g. Could we decide that all these harmonic oscillators could only be excited
up to their 7th level, after that it would not be possible to put more π+ in with a
given p ?
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13 Second Quantization as Cross Product 225
We could of course postulate such a “theory” but it would be rather strange
physically. A postulate of only up to 7 particles per p would violate locality
In a big universe particles with the same momentum are so far from each
other that one cannot from locality feel if there are more or less than 7 particles in
the same momentum eigenstate.
If we use x̃ instead of p then locality would be automatic.
If one thinks of a discretized (d-1)-space, i.e. really a (perhaps a bit irregular)
lattice, and take the state of the universe to be described by the a state in the
Hilbert spaceH, then factorization of the type
H =
⊗
~x
H~x (13.4)
i.e. where we as “factorization parameter” use the spatial position ~x - the lat-
tice point, if discretized - this Cartesian product would be automatically suited
for locality, one should just only provide it with local interaction, but could for
the structure and operators acting on the single factors H~x be very free since
everything would be o.k..
Usual second quantization for the Norma’s spin-charge-family theory
Once one has decided on the inner degrees of freedom, the statistics – fermion
or boson – and of dimension of space time and thus of the dimension of the
momentum vectors, one would than think that there is only one way to second
quantize.
This way will then turn out in the boson case to indeed be of the form that the
full second quantized Hilbert spaceH takes the product form, and thus be written
in the product way.
However, if one starts by a product form and has not gotten it via the stan-
dard procedure, then we would feel a priori unsafe if this would be a physically
meaningful way or not.
It probably depends strongly on the details.
A couple of trivialities on component numbers
i. A Dirac (rather Weyl) massless spinor in an even number d of space time
dimensions has 2
d
2
−1 components.
ii. In Norma’s spin-charge-family theory ([3] and the references therein) there is
not only the usual Dirac spin index with 2
d
2
−1 components, but a quite analogous
family index again with the 2
d
2
−1 components. So in this model the number of
components could be marked by two Dirac indices, or instead using another but
equivalent formalism with projection and nilpotent “operators”. But in any case
of these two formalisms the number of components for a full fermion particle
is the square of the number for an ordinary Dirac construction. The number of
components is therefore 2d−2. One can learn in Ref. [4,5] in this proceedings that:
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226 N.S. Mankoč Borštnik and H.B.F. Nielsen
a. Only operators of an odd character can offer the second quantization
fermions.
b. The operators of an odd character split into two parts, Hermitian conjugated
to each other.
iii. If we ignore momentum and look at one single momentum only, then the
number of different states one could produce by having for this single momentum
various possible numbers with the 2d−2 different components filled or unfilled
would be 22
d−2
. Let us add that the rest of possibilities belong to either the Her-
mitian conjugated partners or have the evenness Character and do not fulfil the
anticommutation relations for fermions (and probably even not for bosons. In any
case the number is much much more than the number of components.
Standard second quantization procedure in factor language
Before telling this standard procedure of quantizing fermions by the factoriza-
tion into the Cartesian product of “subHilbert” spaces, we have to admit that one
cannot do that without some essential modification, which we though postpone to
discuss below in the section called “The problem of fermions”.
However, we are for the moment interested in reaching to the point, where we
can see the problems when one attempts to make a new way of second quantizing
by postulating some algebraic structure for the operators acting on the “subHilbert”
spaces Hp going into the Cartesian product. For this problem presumably the
statistics being fermion or boson statistics may however not matter so much, so
our postponing is not so crucial for that.
iv. Let us first look for a fixed momentum p and calculate which states are
needed to describe the possibilities for filling with the allowed number of particles
(up to one for fermions, and up to infinity for bosons) all the internal states.
v. Then we construct the Hilbert space Hp,of which is just the number of
different ways of filling particles into the different combinations of internal states.
vi. Then finally you can take the Cartesian product and get the genuine Hilbert
space for the full second quantized theory.
Standard way dim(Hp) = 22
d−2
for Norma’s theory.
Since there are (2
d
2
−2)2 component combinations, namely say 2
d
2
−2 genuine
Dirac components, and 2
d
2
−2 family index values, there for assumed fermion-
statistic 22
d−2
possibilities for filling or not filling these 2d−2 difference internal
states.
Thus the Hilbert-space for only one momentum should have the dimension
dim(Hp) = 22
d−2
. (13.5)
(Notice that this space Hp thus has a much bigger dimension than the space of single
particle internal states, which has only dimension = 2d−2 . )
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13 Second Quantization as Cross Product 227
We ignored at first equations of motion.
We have to modify the above simplified proposal by:
vii. Notice that using the momentum energy relation
E2 − ~p2 = 0 (13.6)
we have for each (d-1)-momentum ~p two values for the energy E of the particle,
so that we should let, as already mentioned, as a possibility
p = (~p, E) , (13.7)
meaning a doubling of the space of momenta to be used.
viii. Let us take into account that the (free) equation of motion (the Dirac
equation, the Weyl equation indeed) for a choice of energy E = ±
√
~p2 only allow
a subspace of the internal space of states for the (single) particle,
(/p)ψ = 0 . (13.8)
Standard second quantization as product over (~p, sign(E)).
Letting an index emr denote that we have restricted the single particle sates to
the states obeying the equations of motion (emr = “equation of motion restricted”)
we write the true standard second quantized Hilbert space
Hemr =
⊗
(~p,sign(E))
H(~p,sign(E)),emr, (13.9)
where nowH(~p,sign(E)),emr is constructed from space of single particle internal
states obeying the Dirac equation and having E = sign(E)
√
~p2, which because
of the restriction by the equation of motion has only half the dimensionality of
2d/2−1 in the simple Dirac case or half of 2d−2 in the case with families. So
dim(H(~p,sign(E)),emr) = 22
d−1/2 = 22
d−2
. (13.10)
13.2 The problem of fermions
Yet a problem for Cartesian product form for fermions.
For just constructing the Hilbert space we could claim that this Cartesian
product procedure is o.k. even for fermions, but for the creation and annihilation
operators or the field operators for fermions there is a problem more:
If we take a true Cartesian product and let it be understood that the creation
and annihilation operators for a state with (~p, sign(E)) = p alone shall act on the
Cartesian product factorHp, then we cannot make such fermion creation or annihilation
operators for different p anticommute! Operators acting alone on different Cartesian
product factors will namely always commute.
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228 N.S. Mankoč Borštnik and H.B.F. Nielsen
Suggested trick to solve the anticommutation problem:
Use operators (−1)Fp , where Fp is the fermion number for the fermions in the
Cartesian factorHp.
That is to say to construct the “true creation or annihilation operators” –
b†(i;p) or b(i;p) – for the p Cartesian factor we modify the truly “local ones”,
c†(i;p) and c(i;p) defined so as to only act on the Cartesian factorHp, not touching
the other factors, by multiplying it with a lot of factors of the form (−1)Fp ′ .
Associate in fact to each essentially momentum p a subset of this kind of
essential momenta B(p) and define
b†(i;p) =
 ∏
p ′∈B(p)
(−1)Fp ′
 c†(i;p) (13.11)
b(i;p) =
 ∏
p ′∈B(p)
(−1)Fp ′
 c(i;p) (13.12)
13.3 Dream of Algebra
Although we for fermions must introduce the modification from c†(i;p) to b†(i;p)
in order to achieve the anticommutativity of the annihilation operators b(i;p),
when we build up the Hilbert space construction from a Cartesian product, we
might dream of using this Cartesian product idea to make a generalization of the
algebra for the operators acting on one of these Hilbert spacesHp (we could call
them factor-Hilbert spaces) from which the Cartesian product is made up to a
more general algebra, say F. That is to say we imagine an algebra F consisting of
operators acting on the Hilbert spaceHp.
We can easily think of e.g. a couple operators/elements f, g ∈ F, which e.g.
anticommute {f, g}+ = 0. Of course we shall then have such algebra elements
for every factor-Hilbert-spaceHp, and correspondingly we should of course dis-
tinguish analogous algebra elements related to different factor-Hilbert-spaces or
equivalently different p as we decided to enumerate these factor-Hilbert-spaces.
That is to say we should write f(p) for the operator of a given structure in F when
it acts onHp.
But now if we do not even make the modification of inserting the (−1)Fp ′ -
factors when in the ordering we had to have p and p ′ were in a certain relative
order - say p ′ < p - then of course any f(p) and any g(p) at one p will commute
with any f(p ′) and any g(p ′) at another “momentum” p ′ 6= p, independent of
how f and g for the same p may happen to commute or anticommute.
In other words we cannot prevent the commutation due to independent
factor-Hilbert-spaces for the operators, what ever we take the local algebra to
be, i.e. it does not modify this commutation to let the operators say anticommute
locally, it does not help even if say {f(p), g(p)}+ = 0 to prevent [f(p), g(p ′)] = 0
for p 6= p ′.
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13 Second Quantization as Cross Product 229
13.3.1 Even with (−1)Fp -factors
Even if we improve our purely Cartesian product construction with the (−1)Fp -
factors as above, it will not bring us to get the commutation or anticommutation to
progress from the “local” to the inter p commutator or anticommutator so easily.
If we indeed include the type of factor (from (13.11,13.12) ) being the product over
the factors (−1)Fp ′ for all p ′ which are say “smaller” in the ordering than the p con-
sidered, then we will achieve that we get anticommutation all operators g(p) say
at p with all the ones at another place p ′ provided both operators carry a fermion
number in the sense that they shift the value of the fermion number Fp for their
factor Hilbert space by their action. So if e.g. two operators are fermionic in this
quantum number F sense and even if they commuted when at the same site, they
will anticommute when they are at different sites. If oppositely they anticommute
locally they will again anticommute when at different sites(= different p’s).
The conclusion from the remarks just above should be:
Using the starting point of the Cartesian product and only modifying by
the extra factor of the type from equations (13.11,13.12) the commutation versus
anticommutation of operators associated with different p-values depend alone
on:
a. the fermion number of the operators,
b. from whether one introduce transformation (13.11, 13.12) above at all or not.
But it does not depend on on how the algebra elements considered may commute
or not in the “local algebra” i.e. for the same p-value.
13.3.2 More generally:
The above proposed method for making fermion-fields on the basis of a Cartesian
product by means of an ordering of all the p-values is really not very attractive.
In fact such an ordering does not match well with the topological structure of
a momentum space or a position space except for the spatial dimension being
dspatial = 1. In higher dimensions you rather have to use the axiom of choice to
even see that there exists such an ordering. We also need such a construction if we
would like to make fermionization, and then this only by axiom of choice found
ordering would not seem attractive at all either.
So attempting to generalize this method of constructing fermion fields from a
Cartesian product is highly called for.
Now if there is in the theory some sort of gauge freedom one might not re-
quire quite as strict the properties of the extra factors introduced to convert the
a priori commuting fields appearing from operators acting on different factor-
Hilbert-spaces from (13.11.13.12). If one allows more freedom in the construc-
tion then one might optimistically hope to construct such factors to convert the
boson-commuting operators into fermion ones to have some continuity and thus
compatibility with the topology of a higher dimensional space(than just dimension
=1).
We here at first write down the type of transformation to be made to construct
fermions from commuting fields in a general way. Then one may investigate how
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230 N.S. Mankoč Borštnik and H.B.F. Nielsen
much one needs to require about the multiplying factors U(p,p ′) converting the
bosons to fermions so to speak.
Unfortunately we have not come far in developing these conditions, but just
the thought of looking at it more generally might turn out useful:
b†(i;p) =
 ∏
p ′∈B(p)
U(p,p ′)†
 c†(i;p)
b(i;p) =
 ∏
p ′∈B(p)
U(p,p ′)
 c(i;p)
Not even crudely local b†(~x) unless the modification by U(~x,~x ′) inessen-
tial.
So there should preferably be a “gauge” transformation which could be the
effect of the modification U(~x,~x ′) or “jump over correction”-replacement.
Natural that the U(~x,~x ′) depends on the direction from ~x to ~x ′, and thus is a
function of a point on thee sphere Sd−2.
Also the ‘gauge”like modifications must lie in a groupG. So need map Sd−2 →
G.
13.3.3 Anyons
To exercise constructing other statistics than bosons from the Cartesian product one
would of course like to exercise with two spatial dimensions because this is the first
case after the one spatial dimension case in which there are essentially no problem
and fermionization is already well done. But now just 2 spatial dimensions is the
interesting case in which also Leinaas Myhrheim or anyon statistics is possible[6].
With the suspicion of the gauge symmetries being important in allowing a
more developed choice of the conversion factors U(p,p ′) a first exercise might
be to even construct a system of anyons or first just a pair by electromagnetic
ingredients.
Acknowledgement
The author N.S.M.B. thanks Department of Physics, FMF, University of Ljubljana,
Society of Mathematicians, Physicists and Astronomers of Slovenia, for supporting
the research on the spin-charge-family theory, the author H.B.N. thanks the Niels
Bohr Institute for being allowed to staying as emeritus, both authors thank DMFA
and Matjaž Breskvar of Beyond Semiconductor for donations, in particular for
sponsoring the annual workshops entitled ”What comes beyond the standard
models” at Bled.
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13 Second Quantization as Cross Product 231
Fig. 13.1. Anyons as electric magnetic made.
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