Eight Types of Symmetrically Distinct Vectorlike Physical Quantities

Abstract

The Letter draws the attention to the spatiotemporal symmetry of various vectorlike physical quantities. The symmetry is specified by their invariance under the action of symmetry operations of the nonrelativistic space-time rotation group O(3)×{1,1^{'}}=O^{'}(3), where 1^{'} is a time-reversal operation, the symbol×stands for the group direct product, and O(3) is a group of proper and improper rotations. It is argued that along with the canonical polar vector, there are another seven symmetrically distinct classes of stationary physical quantities, which can be-and often are-denoted as standard three-component vectors, even though they do not transform as a static polar vector under all operations of O^{'}(3). The octet of symmetrically distinct "directional quantities" can be exemplified by two kinds of polar vectors (electric dipole moment P and magnetic toroidal moment T), two kinds of axial vectors (magnetization M and electric toroidal moment G), two kinds of chiral "bidirectors" C and F (associated with the so-called true and false chirality, respectively) and still another two bidirectors N and L, achiral ones, transforming as the nematic liquid crystal order parameter and as the antiferromagnetic order parameter of the hematite crystal α-Fe_{2}O_{3}, respectively.

Full text

arXiv:1312.0548v2 [physics.class-ph] 17 Feb 2014
Eight types of physical ”arrows” distinguished by Newtonian space-time symmetry.
J. Hlinka
Institute of Physics, Academy of Sciences of the Czech Republic, Prague, Czech Republic
(Dated: February 18, 2014)
The paper draws the attention to the spatiotemporal symmetry of various vector-like physical
quantities. The symmetry is specified by their invariance under the action of symmetry operations
of the Opechowski nonrelativistic space-time rotation group O(3) N{1,1′}=O′(3), where 1′is
time-reversal operation. It is argued that along with the canonical polar vector, there are another
7 symmetrically distinct classes of stationary physical quantities, which can be – and often are –
denoted as standard three-components vectors, even though they do not transform as a static polar
vector under all operations of O′(3). The octet of symmetrically distinct ”directional quantities”
can be exemplified by: two kinds of polar vectors (electric dipole moment Pand magnetic toroidal
moment T), two kinds of axial vectors (magnetization Mand electric toroidal moment G), two
kinds of chiral ”bi-directors” Cand F(associated with the so-called true and false chirality, resp.)
and still another two achiral ”bi-directors” Nand L, transforming as the nematic liquid crystal
order parameter and as the antiferromagnetic order parameter of the hematite crystal α-Fe2O3,
respectively.
PACS numbers: 61.50.Ah,11.30.Qc,75.10.-b, 75.25.-j,11.30.Rd
Physical quantities defined by a magnitude and an ori-
ented axis in 3D space are often represented by three-
component Euclidean vectors. Frequently, polar and
axial (or pseudo-) vectors are distinguished, depending
on whether they change their sense or not, respectively,
upon the operation of spatial inversion (parity operation
¯
1).[1–4] For classification of temporal processes or mag-
netic phenomena of vectorial nature, the action of the
time-inversion operator (1′) can be used. For example,
magnetization Mand magnetic field vector Hare ”time-
odd axial” vectors (preserved by ¯
1 operation but chang-
ing their sign under the 1′operation), electric polariza-
tion Por electric field Eare ”time-even polar” vectors,
while other quantities like velocity vor toroidal moment
Tare ”time-odd polar” vectors.[1–5, 20] The two inver-
sion operations generate an Abelian group of 4 elements
{1,¯
1,1′,¯
1′}and 4 one-dimensional irreducible represen-
tations; the symmetry operations this group allows to
classify these vectors into 4 categories (see Table I).[1–4]
The aim of this paper is to emphasize that there are
another four types of quantities, which are also defined by
a magnitude, an axis and a geometrical sign, and which
are also often associated by 3D vectors, but which pos-
sess a different spatio-temporal symmetry than the ex-
amples given in Table I). We are going to specify here
all 8 types of ”directional quantities” (i) by describing
1¯
11′¯
1′vectorial quantity symbol
1 1 1 1 electric toroidal moment G
1-1 1 -1 electric dipole moment P
11 -1 -1 magnetic dipole moment M
1-1 -1 1 (magnetic) toroidal moment T
TABLE I: Action of space (¯
1) and time (1′) inversion oper-
ations on selected examples of vectorial quantities: 1 stands
for the invariance, -1 stands for the sign reversal.[3, 7, 10]
G
P M T N C L
F
FIG. 1: Pictographs of 8 kinds of quantities defined by a sign,
a magnitude and an axis. Letter symbols allow to identify
each pictograph with the symmetry assignment given in Ta-
bles I-III. A rrows in pictographs drawn by dashed lines should
be considered as indicating a stationary current or motion
(time inversion operation does change their sense), while those
in pictographs drawn by full lines are time-irreversible (as e.g.
the electric polarization). Pictographs were inspired by pic-
tures employed for similar purpose in Refs. 5, 8, 9.
their transformation properties under the action of the
elements of the Opechowski general space-time rotation
group O(3) N{1,1′}=O(3).1′=O′(3), (ii) by enumer-
ating the associated limiting groups defining their sym-
metry invariance, (iii) by providing several examples to
each case. We shall also briefly discuss possibilities and
difficulties with introduction of formal algebraic manipu-
lations. Simultaneous considerations about all 8 different
types of such directional quantities can be useful in var-
ious areas of physics.
Basic symmetry argument. These 8 symmetrically
different species are resumed pictographically in Fig. 1.
Why do we have just 8 of such quantities? Let us con-
sider any stationary physical quantity X(attached to
a physical object), which simultaneously defines a two-
valued, geometry-related sign, a nonnegative magnitude
and a unique 1D linear subspace of 3D Euclidean space
(an axis of this quantity), but nothing more. Since the
quantity Xdefines a unique axis in the space, the symme-

2
irr. repr. E¯
1mk2⊥1′¯
1′m′
k2′
⊥symbol
∞ ∞ ∞′∞′
2km⊥2′
km′
⊥
A1g(Σ+
g) 1 1 1 1 1 1 1 1 N
A2g(Σ−
g)1 1 -1 -1 1 1 -1 -1 G
A1u(Σ+
u)1 -1 1 -1 1 -1 1 -1 P
A2u(Σ−
u)1 -1 -1 1 1 -1 -1 1 C
mA1g(mΣ+
g)1 1 1 1 -1 -1 -1 -1 L
mA2g(mΣ−
g)1 1 -1 -1 -1 -1 1 1 M
mA1u(mΣ+
u)1 -1 1 -1 -1 1 -1 1 T
mA2u(mΣ−
u)1 -1 -1 1 -1 1 1 -1 F
TABLE II: Characters of one-dimensional irreducible repre-
sentations for selected elements of ∞/mm.1′(D′
∞h) group.
The extra dash symbol identifies the operations combined
with time inversion. Irreducible representations are labeled
similarly as that of the ∞/mm group[6, 12], the ”m” symbol
indicates antisymmetry with respect to the time inversion,
similarly as it is adopted for grey[13] symmetry groups of
magnetically ordered crystals[14–16].
try of Xcan be classified by those O(3).1′group opera-
tions that leave this axis invariant. Such operations form
an infinite subgroup of O(3).1′that could be denoted as
the ∞/mm.1′or D′
∞hgroup.[5, 10, 11] Moreover, it is
natural to postulate that the magnitude of X(|X| ≥ 0)
does not change under the operations of O(3).1′group.
This implies that transformation properties of Xcan be
fully defined by specifying how its sign is changed when
elements of ∞/mm.1′are applied to it. Since we restrict
ourself only to the quantities for which the sign of X
can have only one of the two possible values, the sym-
metry operation can either preserve the sign or change
it to the opposite one. In other words, the action of the
associated ∞/mm.1′group operations consist in multi-
plication of the geometrical sign of Xeither by 1 or by
-1. In terms of theory of groups, this implies that X
transforms as one-dimensional (necessarily irreducible)
representation of the associated ∞/mm.1′group. It
is known that the ∞/mm (D∞h) group has 4 distinct
one-dimensional irreducible representations[6, 12] so the
∞/mm.1′(D∞hN1,1′) one has twice as much of them.
Therefore, the physical quantities defined by a sign, a
magnitude and an axis can be classified in 8 symmetri-
cally different categories.
Classification by irreducible representations
and basic examples. The list of all one-dimensional ir-
reducible representations of the ∞/mm.1′group is given
in Table II. First column gives the irreducible represen-
tations label following the convention used e.g. in Refs. 6
and 12, resp., the last column contains a letter symbol
used in Table III. and in Fig. 1. Remaining columns in
the table are associated with the classes of symmetry el-
ements of the ∞/mm.1′group. There are various phys-
ical quantities having the listed transformation proper-
ties. For example, polarization (P) and magnetization
(M) transform as A1u(Σ+
u) and mA2g(mΣ−
g) irreducible
¯
11′mklimiting group
Gtime-even axial 1 1 −1∞/m.1′
Ptime-even polar −1 1 1 ∞m.1′
Mtime-odd axial 1 −1−1∞/mm′
Ttime-odd polar −1−1 1 ∞/m′m
Ntime-even neutral 1 1 1 ∞/mm.1′
Ctime-even chiral −1 1 −1∞2.1′
Ltime-odd neutral 1 −1 1 ∞/mm
Ftime-odd chiral −1−1−1∞/m′m′
TABLE III: List of 8 symmetrically distinct ”arrow” quan-
tities and their transformation under three independent op-
erations ∞/mm.1′(D′
∞h) group attached to the axis. (mk
stands for any mirror plane operation parallel to the axis.)
representations, resp. The symbol Tinvokes the often
discussed toroidisation or toroidal moment,[17–21] even
though there are many other more frequently used quan-
tities that also transform as the mA1u(mΣ+
u) represen-
tation, such as electric current, momentum or velocity
of a particle, vector potential or the Poynting vector
S=E×H. It is clear from Table II that this ”magnetic”
toroidal moment Thas a different symmetry than the
”electric” toroidal moment G, the latter exploited e.g.
for characterization of electric polarization vortex states
of small ferroelectric particles[22–24] or poloidal spin
currents[25]. Recently, spontaneous magnetic toroidiza-
tion has been found e.g. in Ba2CoGe2O7crystal[26],
the G-type distortion has been identified e.g. in the
”ferroaxial” structures of CaMn7O12 and RbFe(MoO4)2
crystals.[27–29]
Let us note that Gand Mare symmetric with re-
spect to the perpendicular mirror plane operation m⊥
and Pand Tare symmetric with respect to the paral-
lel mirror plane mk. Thus, none of these quantities is
chiral.[30] In fact, only two irreducible representations
from the Table II. fulfill the group theoretical condition
of a chiral object (absence of improper rotation symme-
try, such as center of inversion or mirror planes[30]): A2u
and mA2u. They are naturally suitable for representa-
tion of chiral directional quantities, as their geometrical
sign can reflect the sign of their enantiomorphism. For
example, a helix might be characterized by its axis, the
magnitude (given by the pitch of the helix) and a geomet-
rical sign, indicating whether the helix is right-handed
or left-handed. Such a chiral quantity Ctransforms as
A2uirreducible representation. As a beautiful example of
mA2uquantity (F) can be taken the antiferromagnetic
order parameter of the linear-magnetoelectric chromite
crystal Cr2O5.[31, 32] This latter kind of chirality, re-
versible upon time reversal, is sometimes called ”false
chirality”.[30, 34]
Finally, there are also two irreducible representations
symmetric with respect to both mkand m⊥(Land N).
The time-odd variant (L) can be used to describe another
type of directional antiferromagnetic order parameter,
e.g. in the hematite crystal α-Fe2O3.[32] The fully sym-

3
metric (A1g) representation is perhaps the most singular
one. It can be associated with the so-called director,
exploited in the theory of liquid crystals to characterize
the spontaneously parallel spatial orientation of rod-like
molecules in nematic phases.[35] In this particular case
there is no reason to define its geometrical sign. However,
there are other N-like quantities that do have a sign. For
example, a consistently defined Frank vector of a wedge
disclination[36–38] should allow to distinguish whether
the disclination can be formed by removing or inserting
material body adjacent to the plane of the cut. At the
same time, this disclination itself is invariant against all
operations of the ∞/mm.1′(D′
∞h) group.[36]
Classification in terms of symmetry invariance
groups. Table II defines fully transformation properties
of various uniaxial quantities discussed above. For many
purposes, it is enough to consider only those symmetry
operations, which leave the quantity invariant.[10, 11, 39]
Such operations form infinite subgroups of the ∞/mm.1′
group. They are listed for each irreducible representa-
tion in Table III. The content of these invariance group
can be easily figured out from the pictographic symbols
in Fig. 1. In addition, each pictograph shows a segment
indicating the magnitude of the quantity and an arrow
associated with its geometric sign (see Figs. 1 and 2).
Arrows in pictographs drawn by dashed lines should be
considered as indicating a stationary current or motion
(time inversion operation does change their sense). This
is the case of time-odd quantities (L, M, T, F). In con-
trast, the arrows in pictographs drawn by full lines should
be considered as time-irreversible (time inversion opera-
tion does not change them, as they have a grey-group[13]
symmetry). These pictographs stands for the time-even
quantities N, G, P, C. Let us note that the P, T, N, L
quantities, symmetric with respect to the parallel mirror
plane operation mk, have arrows only in the radial direc-
tion, while mk-antisymmetric quantities, G, M, C, F,
have all only tangential arrows (bend arrows should be
understood as drawn on a visible curved surface of a co-
axial circular cylinder.) One can also easily distinguish
the single-arrow pictographs of 2⊥-antisymmetric quan-
tities G, P, M, T (proper vectors) from all the double-
arrow graphical symbols standing for 2⊥-invariant quan-
tities N, C, L, F, which we call here as bi-directors.
Meaning of the geometric parity signs, bi-
directors. The fact that the parity sign can be rep-
resented in this way emphasizes its geometrical nature.
Obviously, the strict meaning of the parity sign relies on
some convention, too. For example, the vector of electric
dipole moment is taken as pointing towards the center of
the positive charge (and not the opposite), the arrow as-
sociated with the velocity of a particle is drawn towards
its future position (and not the opposite), the sense of
the electric current refers normally to the velocity of the
positive charges, and the arrow in the pictograph stand-
ing for magnetic dipole moment is that of the equivalent
positive stationary electric current circulating around the
indicated axis.
¯
11′examples
σ1 1 G.G,T.T,P.P,M.M,∇P, mass, charge
ǫ−1 1 P.G,T.M
τ1−1M.G,T.P, time
µ−1−1T.G,M.P, magnetic monopole
TABLE IV: Four scalar types[4] specified according to their
invariance under space-inversion and time-reversal operations
(time-even scalar σ, time-even pseudoscalar ǫ, time-odd scalar
τand time-odd pseudoscalar µ).
Another set of conventions is needed to facilitate the
algebraic representation of such quantities. Typically, a
polar vector is represented by three coordinates defined
by its scalar-product projections to an oriented set of
three orthonormal basis vectors. It is so practical that
we tend to represent all other quantities in a similar way.
In case of ”true” vector quantities (those of Table I),
such algebraic representation is usually defined through
the time derivatives and vectorial products or equiva-
lent rules. In fact, this representation justify the com-
mon usage of the simple P-arrow pictograph for all other
vector quantities of the Table I. For example, magnetic
moment mof a current turn is defined as a vector per-
pendicular to the turn and directed so that the current
observed from the end of vector menvelops the turn
counterclockwise.[40] Therefore, the pictograph for M(as
well as for Gand T) can be formally replaced by that of
P, even though these quantities actually do have a dif-
ferent symmetry (in fact, Fig. 1 could conveniently serve
as the replacement table). Moreover, this algebraic rep-
resentation allows to calculate any scalar and vectorial
products in the usual way. Interestingly, vectorial prod-
ucts of true vectors are true vectors and scalar products of
true vectors transforms as one of the four possible scalar
species[4] (time-even scalar σ, time-even pseudoscalar ǫ,
time-odd scalar τand time-odd pseudoscalar µ, see Ta-
ble IV).
In case of bi-directors, none of the SO(3) operations
can reverse their geometrical sign. It indicates the fun-
damental difficulty with representation of bi-directors by
three-component algebraic vectors. In fact, each of these
bi-director quantities transforms as an ”antitandem” ar-
rangement of two vectors - as a couple (”dipole”) of two
opposite vectors X1and X2(|X1|=|X2|) arranged on
a common axis at some nonzero distance r21 =r2−r1.
Obviously, Ntransforms as an antitandem of two Pvec-
tors, Cas a antitandem of two Gvectors, Las a Tvec-
tor antitandem, Fas a Mvector antitandem. Therefore,
a bi-director can be represented by a simple ”two-body”
term a12 =X2−X1. Here it is assumed that the symme-
try operations act both on the vectors and their position
- operations that change r21 to the opposite are actually
interchanging the sites 1 and 2. The geometrical par-
ity sign of such antitandem quantities could be denoted
as inward or outward, depending whether the vector X2
is parallel or antiparallel to the vector r21 , and so its

4
G
P M T
N
C
L
F
FIG. 2: Pictographs of same 8 kinds of quantities as in Fig. 1,
but with an opposite sense.
evaluation actually requires to know two quantities at a
time, X2and r21. Having this in mind, a range of alge-
braic operations can be nevertheless easily extended to
all the above vectors and bi-directors. For the sake of
convenience, types of the quantities obtained as vectorial
cross products or as multiplication by a scalar are given
in Table V. Let us also note that from symmetry point of
view, time derivative acts here as multiplication by the
time-odd scalar τ, so that e.g. the time derivative of the
bi-director Ltransforms as the bi-director Nand vice
versa.
Classification of axes and concluding remarks.
In general, a physical object may have a physical property
transforming as one the 8 discussed cases only if its sym-
metry invariance group is a subgroup of the correspond-
ing limiting group. For example, macroscopic magneti-
zation can exist only in crystals belonging to 31 different
Heesch-Shubnikov point groups, that are subgroups of
∞/mm′group.[3, 39, 41] If the axis of the limiting su-
pergroup coincides with the symmetry axis of the object,
it is often named according to the associated property
(ferromagnetic axis, polar axis). Other axes could be
similarly labeled as toroidal, truly-chiral, falsely-chiral,
G-axis, fully-symmetric and so on.
The term vector is sometimes employed to describe
phenomena that have a bi-director symmetry. For ex-
ample, so-called Burgers vector is widely used to char-
acterize screw dislocations, which are obviously non-
polar, truly chiral (C-type) objects. Similarly, the an-
tiferromagnetic vector[31–33] is often used to describe
the falsely-chiral (F-type) antiferroelectric order. On
the contrary, the so-called ”chiral vector” or ”vector
chirality”[42–44] is sometimes used to characterize cyclic
spin arrangements on spin loops, for example in triangu-
lar antiferromagnetic lattices, even if the spin arrange-
ment happens to have toroidal symmetry, which is ”uni-
directorial” but achiral (similarly as spin cycloids and
N´eel domain walls [45]).
Finally, it is well known that axial vector Gcan be rep-
resented as a polar antisymmetric second-order tensor.
The bi-director quantities can be also classified within
the established tensorial calculus.[1, 4, 46, 47] They cor-
respond to a special kind of second rank tensors, that has
been once coined in Russian literature as the ”simplest
tensor” (i.e. symmetrical second rank tensor having in
its canonical form only a single nonzero element).[8] In
A∼G P M T N C L F
[G×A] or [σA]∼G P M T N C L F
[P×A] or [ǫA]∼P G T M C N F L
[M×A] or [τA]∼M T G P L F N C
[T×A] or [µA]∼T M P G F L C N
TABLE V: Look-up table of transformation properties of vec-
torial products and scalar multiplications. The symbol ∼has
a meaning of ”transforms as...”, the operations involving bi-
director quantities N,C,L,Fare defined in the text.
particular, N-type bi-director could be considered as dual
to the simplest time-even polar tensor, C-type bi-director
transforms as the simplest time-even axial tensor, L-type
bi-director as the simplest time-odd polar tensor and F-
type bi-director as the simplest time-odd axial tensor.
Nevertheless, we think that the unifying classification
via irreducible representations of ∞/m.1′still provides
a very practical concept, applicable in various areas of
physics. In solid state physics at least, the simple per-
spective where vectors and bi-directors have equal le-
gitimacy might be useful when dealing with problems
where several such quantities are interacting, for classifi-
cation of long-wavelength excitations or structural com-
ponents of magnetoelectric multiferroic crystals[11, 48–
50], for description of macroscopic properties of chiral
objects,[51, 52] or for classification of topological defects
like domain walls, magnetic vortices or skyrmions.[53] In
fact, we would like to offer a more complete discussion of
possible applications of this concept in future and so we
would be grateful to learn about other cases where this
perspective could bring some useful insight.
Acknowledgments
It is a real pleasure to acknowledge V´aclav Janovec for
his valuable suggestions related to this work, financially
supported by the Czech Science Foundation (Project
GACR 13-15110S).
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