COMPARATIVE STUDY OF GEOMETRIC PRODUCT AND MIXED PRODUCT:

Md. Shah Alamand 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. 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. 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. APPLICATIONS OF THESE PRODUCTS IN DEALING WITH DIFFERENTIAL OPERATORS

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)

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. 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
matrix algebra / It is not directly consistent
with Pauli matrix algebra / It is directly consistent with
Pauli matrix algebra
3.Consistency with Dirac
equation / It is not consistent with
Dirac equation / It is consistent with Dirac
equation
4. In dealing with
differential operators / It can not be used in dealing
with differential operators / It can be used successfully
in dealing with differential
operators
5. Elementary properties / (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. / (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.

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.

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 [

[5] Md. Shah Alam, Comparative study of Quaternions and Mixed Number, Journal of Theoretics, Vol-3, No-6. 2001 [

[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[

[9] A. kyrala, Theoretical Physics, W.B. Saunders Company, Philagelphia & London, Toppan Company Limited, Tokyo, Japan.

[10]

[11]

[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.

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