DSpace at VNU: Method for the partial wave scattering problem for the quantum field theory

Method for the partial wave scattering problem for the quantum field theory Nguyen Dinh Thinh Hanoi University of Science, VNU; Faculty of Physics Major: Theoretical Physics - Mathemat

Method for the partial wave scattering problem for the

quantum field theory Nguyen Dinh Thinh

Hanoi University of Science, VNU; Faculty of Physics Major: Theoretical Physics - Mathematical Physics

Code: 60 44 01 Supervisors: Prof.PhD Nguyen Xuan Han

Date of Presenting Thesis: 2011

Abstract. Nghiên cứu các phương pha ́p giải phương trình Schrodinger trong

cơ ho ̣c lượng tử: phương pháp khai triển theo sóng riêng phần; phương pháp hàm Green; phương pháp chuẩn cổ điển; mối liên hệ giữa biên độ tán xạ theo sóng riêng phần và biên độ tán xạ eikonal Trình bày sơ đồ mối liên hệ giữa các phương pháp của bài toán tán xạ Phân tích các hiê ̣u ứng hấp dẫn và điê ̣n từ trong bài toán tán xa ̣ ở năng lượng Plangck như tán xa ̣ toàn phần toàn phần hấp dẫn; cực điểm của tán xạ; tán xạ hấp dẫn có kể thêm tương tác điện từ.

Keywords. Sóng; Vật lý lý thuyết; Tán xạ; Trường lượng tử; Vật lý toán

Content

In recent years there have been important advances in our understanding of scattering at the Planck scale energy in quantum field theory / 1-10 / To study this process in the theory of quantum gravity will provide a scientific basis to be aware of physical phenomena such as the birth of the singularity and the formation of black holes, the loss of information as well as the improvement variable's string theory of gravity The results obtained are confirmed Planck scattering amplitude of high-energy particles in the size (where s is the energy of the hat, is the Planck mass, - is the gravitational constant) and t-squared pulse of transmission is small, within the limits of the form eikonal representation - representation Glauber (leading term) phase depends on energy Additional terms of (non-leading terms) in the scattering problem has been many domestic and foreign scientific research interest over 20 years, including the Department

Department of Physics theory is to find the first-order terms added to the terms

of the amplitude eikonal scattering amplitude in the theory of quantum gravity, using both methods are different methods of analysis distributed gossip content and the standard equation / 8-9 / Finding other methods for this problem is still topical issues

- It offers three methods of solving the Schrodinger equation to find the scattering amplitude in which the partial wave method A comparison of three methods to help us have different directions for the scattering problem in quantum mechanics

- Method of partial waves used in quantum mechanics is generalized, then it

is used to study the scattering problem in the Planck energy theory of quantum gravity

I The Schrodinger equation method

1.Phuong method developed by the partial wave Schrodinger equation:

Total wave function describing the motion of the

2 2

h

particle toparticle scattering and at large distances (r> a) for interest equal to the sum of scattering and wave to the scattered wave:

ikr ikz e

r

r

Return to R equations obtained radial equation of the form:

2

1

0

dr dr

scattering amplitude in partial waves

2

1

2

l

i

ik

d

2 Green function method

Schrodinger equation:

( ) ( ) ( )

,

Equation can be rewritten as integral equation:

3 0

( )r ( )r d r G r r U r' ( , ') ( ') ( ')r

,

Under the boundary conditions, the wave function must includetwo components: component waves to the plane wave traveling inthe

positive z axis and the rest is scattered spherical wave Sorewritten as:

'

0

1

ik r r

i k r e

r r

-r -r

r r

r r

Scattering amplitude in partial waves

0 0

i

¥

c

3 The standard method of classical

Also derived from the Schrodinger equation (and test of the equation of the form:

y = eiS(x)/h

So the Schrodinger equation we obtain:

h

2

2m i S + 2m S = E - U

h

In the classical limit, and instead we have:

'( )

( )

U x

-h

Integral expression

2

2

z L

Derived from the wave function of the form:

2 2

2 ( ' ) ' 3/ 2

1 (2 )

z L

im

U b z dz k

ikz

e e

y =

p

h

(

The scattering amplitude is written:

2 '

2

1 2

4

ik x ikx z

L

m

im

k

p

ò ò

h h

The amplitude of scattering is calculated according to standardclassical

0 0

i

¥

c

4 Contact between the scattering amplitude in partial wave and

the eikonal scattering amplitude

l

l 0

1

2ik

d

¥

=

l

2i ( k) l

0

1

2ik

d

¥

With the problem of high energy scattering, is considered to belarge we can replace the summation by the integral l l

When the angle is small, we have:

+ ççè ø÷÷= + ççè ø÷÷= ççè ø÷÷

II The gravitational effects

1 Gravitational scattering completely

Starting from the general covariant equation Klein-Gordon formassless

1

g

m

particles - such as nuclear "test" in the gravitational field and electromagnetic field:

,

mn mn

-, A xm( ) as the electromagnetic field

First we consider the gravitational scattering completely, that is,consider the scattering of neutral particles So we set in

Where the classical Schwarzschild beer slow motion of the particle (the

particle mass M is considered to be small beercompared to) obtained by

the experiment of Einstein equations,the form:

1

= - ççç - ÷÷ + ççç - ÷÷ + q + q f

energy value is the center of mass is very strange:

i

N

+

The formula above allows to draw out some of the majorrejection, and has been

function variables associated with increased Gamma inverse exponent of l The result we get

log

l

Gs

d » - ê - ú+ + ç ÷ççè ø÷

2 Culmination of the scattering amplitude

First, we will re-expression eikonal scattering amplitude asobtained in the first chapter:

2 2

0

2

l

i ikb

i s

f s t = ¥ d be ée d - ù

p ò

And with attention Mandelstam variables As such, we will

rewrite the complementary expression of the scatteringamplitude:

1

iGs iGs iGs

÷

3 2

iGs

æ ö

G - ç- ÷÷

G + è ø

1 2

1 2 (1)

1 2

4

iGs iGs iGs iGs

Gs

t iGs

1 2 1 2

s iGs t

æ ö

G - ç- ÷÷

3 Scatter more attractive since the electromagnetic interaction

The first one considers the scattering

of neutral particles inexternal test metric

Reissner-Nordstom by the static charge.Klein Gordon equation for the fast

moving particles can also beobtained by replacing the

Nordstom:

,

1

( , )

i Gs QQ

i Gs QQ

f s t

s

i Gs QQ t

-æ ö

- Method of partial waves used in quantum mechanics isgeneralized, then it is used to study the scattering problem in thePlanck energy theory of

quantum gravity

- Have shown that for neutral particles, the peak of

thescattering amplitude in partial wave method lies on the imaginaryaxis energy-momentum The culmination of this was distributed atlocations other

than where they appear in the eikonalapproximation

- For particles with electric charge, the effects of

electromagnetic and gravitational fields remain separate when using

the eikonal approximation, and

obtained the additionalterms of the momentum transfer The effects of

electromagneticand gravitational disturbances would be together

as additionalprimary research at higher levels

References

I Vietnameses

1 Nguyễn Ngọc Giao (1999), Lý thuyết trường hấp dẫn, Đại học Quốc gia

TPHCM

nội

4 Nguyễn Xuân Hãn (1998), Cơ sở lý thuyết trường lượng tử, ĐHQG Hà

Nội, Hà nội

II English

1 t Hoof, (1988) “On the Factorization of Universal Poles in a Theory of

Gravitating Point Particles”, Nucl Phys B304, pp 867-876

2 D.Amati, M.Ciafaloni and G.Veneziano, (1988) “Classical and Quantum Gravity Effects from Planckian Energy Superstring”, Int J Mod Phys A3,

pp1615-1561

3 H Verlinde and E Verlinde, (1992)” Scattering at Planckian Energies”,

Nucl.Phys.B371, pp 246-252

4 D.Kabat and M Ortiz, (1992) “Eikonal Gravity and Planckian Scattering”,

Nucl.Phys.B388, pp.570-592

5 Nguyen Suan Han and Eap Ponna; (1997) “ Straight-Line Path Approximation for the Studying Planckian-Scattering in Quantum Gravity”,

Nuo Cim A, N110A pp 459-473

6 Nguyen Suan Han, (2000) “Straight-Line Path Approximation for the High-Energy Elastic and Inelastic Scattering in Quantum Gravity” Euro Phys J C, vol.16, N3 p.547-553 Proceedings of the 4th International

Workshop on Graviton and Astrophysics heid in Beijing, from October

10-15, 1999 at the Beijing Normal University, China, Ed Liao Liu, et al World Scientific Singapore (2000)pp.319-333

7 S Das and P Majumdar, (1998) “Aspects of Planckian Scattering Beyon the Eikonal ” Journal Pramana, India, 51, pp 413-418

8 Nguyen Suan Han and Nguyen Nhu Xuan (2002), “Planckian Scattering Beyon the Eikonal Approximation in the Functional Approach” E-print arxiv: gr-qc/0203054, 15 mar 2002, 15p; European physical journal c, Vol

24, pp.643-651

9 Nguyen Suan Han and Nguyen Nhu Xuan, (2008)“ Planckian Scattering Beyon the Eikonal Approximation in the Quasi-Potential Approach” E-print

arxiv: 0804.3432 v2 [quant-ph] To be published in european physical journal c (2008)

10 Rosenfelder r (2008), “Path Integrals for Potential Scattering”,E-print arxiv: 0806.3217v2[nucl-th]

11 Charles Poole Herbert Goldstien and John Safko Classical Mechanics Addison Wesley

12 Robert A Leacock and Michael J Padgett “Hamilton-Jacobi Theory and the Quantum Action Variable” Physical Review Letters, 50(1):3–6, 1983

13 Robert A Leacock and Michael J Padgett “Hamilton-Jacobi/action-angle

quantum mechanics” Physical Review D, 28(10):2491–2502, 1983

14 Marco Roncadelli and L.S Schulman “Quantum Hamilton-Jacobi

Theory”

Physical Review Letters, 99(17), 2007

15 t Hoof, (1988) “On the Factorization of Universal Poles in a Theory of

Gravitating Point Particles”, Nucl Phys B304, pp 867-876

A.K Kapoor R.S Bhalla and P.K Panigrahi “Quantum Hamilton-Jacobi formalism and the bound state spectra” arXiv, quant-ph/9512018v2, 1996

16 J.J Sakurai Modern Quantum Mechanics Pearson Education, 2007