COMPARATIVE STUDY OF GEOMETRIC PRODUCT AND MIXED
PRODUCT:
Md. Shah Alam and M.H. Ahsan
Department of Physics, Shahjalal University of Science and Technology, Sylhet,
Bangladesh.
ABSTRACT
Scalar and vector product of vectors are well known. Here we discussed another types of
product of vectors, such as Geometric product and Mixed product. It was observed that
Mixed product is more consistent with Physics than that of Geometric product.
Key Words: Geometric product, Mixed Product
PACS No: 02.90. + p
1
1. INTRODUCTION
(i) Geometric product:
Bidyat Kumar Datta and his co-workers defined the Geometric product as [1,2]
A B = A.B + A  B
…………………. (1)
Where A and B are two vectors, A  B i.e. A wedge B which is different from the usual
cross-product in the sense that it has magnitude ABsin and shares its skew property
A  B =  B  A, but it is not a scalar or a vector: it is directed area, or bivector, oriented
in the plane containing A and B.
(ii) Mixed product:
Mixed number [3,4,5,6,7,8]  is the sum of a scalar x and a vector A like
quaternion[9,10,11]
i.e.  = x + A
The product of two mixed numbers is defined as
 = (x + A)(y + B) = xy + A.B + xB + yA + iAB
…………………. (2)
Taking x = y = 0 we get from equation (2)
A  B = A.B + iA  B
…………………. (3)
This product is called mixed product and the symbol  is chosen for it.
2
2. CONSISTENCY OF GEOMETRIC PRODUCT AND MIXED PRODUCT
WITH PHYSICS
(i) Consistency with Pauli matrix algebra.
It can be shown that [12]
(.A)(.B) = A.B + i.(A  B)
…………………. (4)
where A and B are two vectors and  is the Pauli matrix. From equation (3) and (4) we
can say that the mixed product is directly consistent with Pauli matrix algebra. From
equation (1) and (4) we can also say that the Geometric product is not directly consistent
with Pauli matrix algebra.
(ii) Consistency with Dirac equation.
Dirac equation (E - .P - m) = 0 can be operated by the Dirac operator (t - .V - n)
then we get
(t - .V - n) {(E - .P - m)} = 0
………………… (5)
For mass-less particles i.e. for m = n = 0 we get [13]
(t - .V)( E - .P) = [{tE + V.P + i.(VP)}1 + {(t.P + E.V)}2 = …………(6)
Where  is the wave function and 1 and 2 are the components of .
Putting t = 0 and E = 0 in the equation (6) we get
(.V)( .P)  = {V.P + i.(VP)}1
…………………. (7)
Therefore from equation (3) and (7) we can say that the mixed product is consistent with
Dirac equation. From equation (1) and (7) we can also say that the Geometric product is
not consistent with Dirac equation.
3
3. APPLICATIONS OF THESE
DIFFERENTIAL OPERATORS
PRODUCTS
IN
DEALING
WITH
In region of space where there is no charge or current, Maxwell’s equation can be written
as
(i) .E = 0
(ii) E =  (B)/(t)
......………….. (8)
(iii) .B = 0
(iv) B = 00(E)/(t)
From these equations it can be written as [14]
2E = 00(2E)/(t2)
…………………… (9)
2B = 00(2B)/(t2)
Using equation (3) and (8) we can write
  E = .E + i  E
= 0 + { i(B)/(t)}
or,   E =  i(B)/(t)
……………………. (10)
or, (  E) =   { i(B)/(t)}
=  i(/t) {  B}
=  i(/t) {.B + i  B}
=  i(/t){ 0 + i 00(E)/(t)}
or, (  E) = 00(2E)/(t2)
…………………… (11)
It can be shown that  (  E) = 2E
.............…………. (12)
From equation (11) and (12) we can write
2E = 00(2E)/(t2)
Which is exactly same as shown in equation (9)
4
Similarly using mixed product it can also be shown that
2B = 00(2B)/(t2)
Therefore mixed product can be used successfully in dealing with differential operators.
Using the definition of Geometric product (equation 1) it can be shown that Geometric
product can not be used in dealing with differential operators.
4. ELEMENTARY PROPERTIES OF THESE PRODUCTS
(1) Elementary properties of Geometric product
(i)
Geometric product of two perpendicular vectors is an area or bivector oriented in
the plane containing the vectors.
(ii)
Geometric product of two parallel vectors is simply the scalar product of the
vectors.
(iii)
It is satisfies the distribution law of multiplication.
(iv)
It is non-associative.
(2) Elementary properties of mixed product
(i)
Mixed product of two perpendicular vectors is equal to the imaginary of the
vector product of the vectors.
(ii)
Mixed product of two parallel vectors is simply the scalar product of the vectors.
(iii)
It is satisfies the distribution law of multiplication.
(iv)
It is associative.
5
5. TABLE: COMPARISION OF GEOMETRIC PRODUCT AND MIXED
PRODUCT
Geometric product
Mixed product
1. Mathematical expression
AB = A.B + A  B
A  B = A.B + iA  B
2. Consistency with Pauli
It is not directly consistent
It is directly consistent with
matrix algebra
with Pauli matrix algebra
Pauli matrix algebra
3.Consistency with Dirac
It is not consistent with
It is consistent with Dirac
equation
Dirac equation
equation
4. In dealing with
It can not be used in dealing
It can be used successfully
differential operators
with differential operators
in dealing with differential
operators
5. Elementary properties
(i) Geometric product of
(i) Mixed product of two
two perpendicular vectors is perpendicular vectors is
an area or bivector oriented
equal to the imaginary of
in the plane containing the
the vector product of the
vectors.
(ii) Geometric product of
vectors.
(ii) Mixed product of two
two parallel vectors is
parallel vectors is simply
simply the scalar product of
the scalar product of the
the vectors.
(iii) It is satisfies the
vectors.
(iii) It is satisfies the
distribution law of
distribution law of
multiplication.
multiplication.
(iv) It is non-associative.
(iv) It is associative.
6
6. CONCLUSION
Mixed product is directly consistent with Pauli matrix algebra and Dirac equation but
Geometric product is not directly consistent with Pauli matrix algebra and Dirac
equation. Mixed product can be used successfully in dealing with differential operators
but Geometric product can not be used in dealing with differential operators. Moreover,
Mixed product is associative and Geometric product is non-associative. Therefore, Mixed
product is more consistent with different laws of Physics than that of Geometric product.
It could be concluded that Mixed product is better than Geometric product.
ACKNOWLEDGEMENT
We are grateful to Mushfiq Ahmad, Dept. Physics, University of Rajshahi, Rajshahi,
Bangladesh for his help and advice.
7
REFERENCES
[1] B.K. Datta, V. De Sabbata and L. Ronchetti, Quantization of gravity in real space
time, IL Nuovo Cimento, Vol. 113B, No.6, 1998
[2] B.K. Datta and Renuka Datta, Einstein field equations in spinor formalism,
Foundations of Physics letters, Vol. 11, No. 1, 1998
[3] Md. Shah Alam, Study of Mixed Number, Proc. Pakistan Acad. Sci. 37(1): 119-122.
2000
[4] Md. Shah Alam, Mixed Product of Vectors, Journal of Theoretics, Vol-3, No-4. 2001
[http://www.journaloftheoretics.com]
[5] Md. Shah Alam, Comparative study of Quaternions and Mixed Number, Journal of
Theoretics, Vol-3, No-6. 2001 [ http://www.journaloftheoretics.com]
[6] Md. Shah Alam, Different types of product of vectors, News Bulletin of the Calcutta
Mathematical Society, Vol. 26. 2003
[7] Md. Shah Alam, Comparative study of mixed product and quaternion product, Indian
Journal of Physic-A, Vol.77, No. 1. 2003
[8] Mushfiq Ahmad and Md. Shah Alam, Extension of Complex Number by Mixed
Number Algebra. Journal of Theoretics, Vol-5, No-3. 2003
[http://www.journaloftheoretics.com]
[9] A. kyrala, Theoretical Physics, W.B. Saunders Company, Philagelphia & London,
Toppan Company Limited, Tokyo, Japan.
[10] http://mathworld.wolfram.com/Quaternion.htm
[11] http://www.cs.appstate.edu/~sjg/class/3110/mathfestalg2000/quaternions1.html
[12] L. I. Schiff, Quantum Mechanics, McGraw Hill International Book Com.
[13] Md. Shah Alam, Shabbir Transformation and its relativistic properties, M. Sc.
Thesis, Department of Physics, University of Rajshahi, Rajshahi, Bangladesh – 1994.
[14] David J. Griffiths, Introduction to Electrodynamics, Second edition, Prentice-Hall of
India Private Limited, New Delhi 1994.
8
9