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The dressed oscillator approach and particle creation in two simple models of a Friedmann-Robertson-Walker… Bruskiewich, Patrick 2001

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The Dressed Oscillator Approach and Particle Creation in Two Simple Models of a Friedmann-Robertson-Walker Universe by Patrick Bruskiewich B.Sc. University of British Columbia, 1984 A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in The Faculty of Graduate Studies Department of Physics and Astronomy We Accept this Thesis as conforming to the Reauired Standard The University of British Columbia April, 2001 © Patrick Bruskiewich 2001 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia • Vancouver, Canada DE-6 (2/88) Abstract In the First part of this thesis I look at the Algebraic Method which is a very straightforward technique. The idea behind the Algebraic Method is to gener-ate all the states of a quantum system beginning with a well defined base state, generally the lowest energy state, through successive application of a creation operator (also known as a raising operator) which modifies the lowest energy state in such a fashion as to then characterize the rest of the spectrum of the system. The lowest energy state is defined as the state that is annihilated by the annihilation operator (also known as the lowering operator). Several examples of the Algebraic technique are presented including Landau Levels. In the Second part of this thesis I look at several examples of Unitary Simi-larity Transformations and how they can be used to simplify Hamiltonians de-scribing quantum systems. Examples of the Similarity Transformation Method discussed in this thesis include a method to determine the ground state eigen-function using a generating function, Electron-Spin Resonance, the Foldy and Wouthuysen Transformation and an approach first proposed by Wentzel and applied by Schwinger to describe the non-relativistic interaction of an electron with a field. Schwinger used this approach to solve for the Lamb shift of the electron in a central coulombic potential. In the Third Part of this thesis I look at the Bogoliubov Transformation which can be used for Diagonalizing a Quadratic Bosonic Hamiltonian. In the Fourth Part I describe the coupling between a non-relativistic system of oscillators coupled linearly to a scalar field in ordinary Euclidean 3-space. From a physical point of view we give a nonperturbative treatment to the os-cillator radiation introducing some coordinates that permit us to divide the coupled system into two parts, the "dressed" oscillator and the field. I also look at how one can describe transitions due to a forcing function. The first four sections of this thesis build up the mathematical tools, namely the Algebraic Method, the Bogoliubov transformation and the "dressed" oscil-lator approach, for Part Five in which I look at uniform acceleration n Rindler space, particle creation in two simple models of a Friedmann-Robertson-Walker Universe, as well as a hypothesis that Gravity is an Induced Quantum Effect. i i Contents Abstract i 1 Acknowledgements vh 1 Introduction 1 1 The Algebraic Method 3 2 Introduction 4 3 Building up an Operator using Creation and Annihilation Op-erators 5 4 The Quantum Harmonic Oscillator 10 5 Solving for the | 0 > state for the Quantum Harmonic Oscillator 13 6 Angular Momentum 15 7 The Rigid Rotator 19 8 Solving for the Lower Bound | I > state of the Rigid Rotator 22 9 Spherical Symmetric Potentials 24 10 A Charged Particle in a Magnetic Field: Landau Levels 33 11 Second Quantization 37 11 Similarity Transformations 40 12 Introduction 41 13 The Similarity Transformation 42 14 The Fundamental Theorem of Algebra Applied to the Q H O 44 i i i 15 A Similarity Transformation Applied to the Q H O 46 16 Formal Expression for the Similarity Transformation 50 17 The Generating Function 53 18 The Hermite-Lindemmann Transcendental Theorem 55 19 The Base State of the Hamiltonian 56 20 Example Hamiltonians 59 21 Describing Electron-Spin Resonance Using A Similarity Trans-formation 67 22 Describing the Interaction of an Electron with a Field 69 23 The Foldy-Wouthuysen Similarity Transformation 74 24 The F W Transformation Applied to an Electron in a Field 76 III The Bogoliubov Transformation 80 25 Introduction 81 26 The Bogoliubov Transformation 82 27 Field Quantization and Spin-Resonance 91 IV The Dressed Oscillator 96 28 Introduction 97 29 A n Exact Approach to Oscillator Radiation using A Contact Transformations 98 30 Transitions Due to a Forcing Function 111 V Cosmological Particle Creation 115 31 Introduction 116 32 Particle Creation in a Two-Dimensional F R W Universe 117 33 Particle Creation in a Four-Dimensional Model Universe 120 iv 34 Uniform Acceleration in Rindler Space 123 35 Is Gravity an Induced Quantum Effect 127 VI Summary and Conclusions 132 36 Summary and Conclusions 133 VII Bibliography 134 VIII Appendices 138 A Some Observable Effects of Zero Point Fluctuation 139 B Zero Point Fluctuations and the Suspended Charge Paradox 145 C Why the Cut-off? 162 V Acknowledgments I would like to acknowledge the ideas and encouragement of the following individuals in preparing this thesis: Dr. M . McMil lan , Dr. J . McKenna, Dr. K . Schleich and Dr. A . Zhitnitsky. I would also like to thank my wife Kris ta for her patience and support. ©2001 Patrick Bruskiewich Chapter 1 Introduction The investigation of most quantum systems leads to the solution of the Schrodinger equation with a well chosen interaction Hamiltonian or potential. Unfortunately, exact solutions for the Schodinger equation are known for a rather restricted set of interaction Hamiltonian or potentials, so the standard problem we are faced with is to find a good approximation in place of an exact solution. Confronted with a challenge of finding an approximate solution, the goal becomes to obtain a representation H = Ho + Hint, m which Ho describes a known physical system with characteristics close to H, and the interaction Hamiltonian Hint provides a correction to the Hamiltonian Ho. To describe a quantum system means choosing a Hilbert space of states on which the canonical variables are defined as operators. In turn, this means that definite representations of the canonical (or anti-) commutation relations has been chosen. For the case of quantum systems with a denumerably finite number of degrees of freedom, all representations are unitary equivalent to each other. This fact may be used to our advantage when attempting to solve the Schrodinger equation for the quantum system. Many of our descriptions of quantum systems have been influenced by the Quantum Harmonic Oscillator (QHO). For quite a wide range of quantum sys-tems, it is valid to look for an initial approximation in the form of an oscillator basis, that is, a stable quantum system in a well chosen representation can be described by some set of harmonic oscillators with a spectrum of frequencies. Many systems may be treated as a set of oscillators with a frequency defined by a mass parameter. The interaction does not change the oscillator nature of the underlying quantum field, but only redefine their masses and other physical characteristics. The use of the Algebraic Method and Unitary Similarity Transformations 1 in quantum mechanics has proven to be useful particularly when dealing with systems with discrete spectrum. In the First part of this thesis I look at the Algebraic Method which is a very straightforward technique. The idea behind the Algebraic Method is to generate all the states of the system beginning with a well defined base state, generally the lowest energy state, through successive application of a creation operator (also known as a raising operator) which modifies the lowest energy state in such a fashion as to then characterize the rest of the spectrum of the system. The lowest energy state is defined as the state that is annihilated by the annihilation operator (also known as the lowering operator). The methods outlined in this part of the thesis are by no means the only methods to char-acterize the eigenfunctions of quantum systems. Other techniques such as the Factorization Method of Hull and Infeld [1] may be used, as well as a number of more specialized techniques. In the Second part of this thesis I look at Unitary Similarity Transformations and how they can be used to simplify Hamiltonians describing quantum systems. Examples of the Similarity Transformation Method discussed in this thesis in-clude a method to determine the ground state eigenfunction using a generating function, Electron-Spin Resonance , the Foldy and Wouthuysen Transformation and an approach first proposed by Wentzel and applied by Schwinger to describe the non-relativistic interaction of an electron with a field. Schwinger used this approach to solve for the Lamb shift of the electron in a central coulombic po-tential. In the Third Part of this thesis I look at the Bogoliubov Transformation which can be used for Diagonalizing a Quadratic Bosonic Hamiltonian. In the Fourth Part I describe the coupling between a non-relativistic system of oscillators coupled linearly to a scalar field in ordinary Euclidean 3-space. From a physical point of view we give a nonperturbative treatment to the os-cillator radiation introducing some coordinates that permit us to divide the coupled system into two parts, the "dressed" oscillator and the field. I also look at how one can describe transitions due to a forcing function. The first four sections of this thesis build up the mathematical tools, namely the Algebraic Method, the Bogoliubov transformation and the "dressed" oscil-lator approach, for Part Five in which I look at uniform acceleration in Rindler space, particle creation in two simple models of expansion in a Friedmann-Robertson-Walker Universe, as well as the hypothesis that gravity is an induced quantum fffect. 2 Part I The Algebraic Method Chapter 2 Introduction The idea behind the Algebraic Method is to generate all the states of the sys-tem beginning with a well defined base state, generally the lowest energy state, through successive application of a creation operator (also known as a raising operator) which modifies the lowest energy state in such a fashion as to then characterize the rest of the system. The lowest energy state is defined as the state that is annihilated by the annihilation (lowering) operator. Key to the application of the Algebraic Method is the proper formulation of the creation operator, the operator algebra and the formulation of the funda-mental state. The Algebraic Method amounts essentially to the replacement of a second-order differential equation by equivalent products of first-order equa-tions from which the appropriate creation operators are identified. The archetypical discrete system that is built up using the Algebraic Method is the Quantum Harmonic Oscillator (QHO) which was first proposed by P . A . M . Dirac early in the development of Quantum Mechanics. [2] The Algebraic Method has also been applied to other discrete quantum sys-tems. The example of a Charged Particle in a Magnetic Field and how one arrives at the Landau Levels is given as an example of the Algebraic Method applied to a degenerate quantum system. 4 Chapter 3 Building up an Operator using Creation and Annihilation Operators Let Pn be a Hermitian operator in Hilbert space with a discrete spectrum and let | n > be a denumerable set of eigenvectors so that (n = 0 ,1 ,2 ,3 , . . . ) . P\n>=pn\n> (3.1) Construct a creation operator and an annihilation operator n associated with P . [3] The creation operator will have the form 7/t = ^2 Cn \ n + l><n\ (3.2) n and the annihilation operator will have the form r] = ^2 °n-i | n - l > < n | (3-3) n Applying these operators to an eigenstate of the operator P yields 7?t | k >=Ck | k + 1 > (3.4) and V\k>= CU \ k - l > (3.5) 5 It is possible then to define the relationship between the coefficients and the creation and annihilation operators in the following fashion, namely 7777* I * >=l Ck | 2 | k > (3.6) and similarly 77*77 | k >=\ C k - i | 2 | k > (3.7) Assume now that the spectrum for P has a lower bound corresponding to n = 0. so that C _ i = 0. This defines a zeroth state | 0 > in such a way that 77 | 0 >= 0. If there exists an upper bound at state N then CN = 0. Not all systems have an upper bound. Since the operators 7777* and 77*77 have the same eigenvectors as P , it is pos-sible to formulate an expression for P in terms of 7777+ and 77*77. Without losing any generality we write P as an ordered function, P = Y, °mntfv)n ( W ) m (3-8) m,n It is now possible, through a wise choice of coefficients Ck to reduce an operator such as a quadratic Hamiltonian into an ordered function of a simple linear form, of the creation and annihilation operators, P = a00 + aioVV^ + aoirfr) (3.9) that is when operating on | n > P = aoo + 010 I Cn I2 +aoi | C„_ i | 2 (3.10) It is convenient for the purpose of analysis to separate out the anti-symmetric and symmetric combinations of the creation and annihilation operators. Define the anti-symmetric A operator by A = 7777* - 77*77 (3.11) and the symmetric operator 5 by S = 7777* + 77*77 (3.12) 6 So then we can express P in terms of the anti-symmetric and symmetric operators P = q0 + qaA + g s 5 (3.13) Let a* and sk be the eigenvalues of the operators A and 5, respectively, so then A | k >= ak | fc >= (| Ck | 2 - | C f c _ ! | 2 ) | fc > (3.14) and S\k>=sk\k>=(\Ck\2+ \ C f c _ ! | 2 ) | > (3.15) yielding for the eigenvalues ak =| Ck | 2 - | C * _ i | 2 (3.16) and sk =| C f c | 2 + | C * _ i | 2 (3.17) From these two equations it follows that I Ck | 2 = i ( « f c + ofc) = i ( * f c + i - a , + 1 ) (3.18) From these results it is possible to set up a series of relations known as consistency relations which prove valuable in the analysis of specific systems. The consistency relations are the following: sk > 0 (3.19) sk + ak > 0 (3.20) sk - ak > 0 (3.21) From the equation for | Ck | 2 we also have 7 Sk+ak = sk+i - ak+i (3.22) which fixes the spectrum of the operator P, which along with the lower bound term C-\ — 0 and the upper bound term — 0 provides the final pair of consistency relations so = a0 (3.23) and if an upper bound indeed exists sjv = —a,N (3-24) This set of consistency relations contains all the information about the spec-trum of the operator P. These relations are compact and easier to use then other techniques, provided one can formulate the operators A and 5 for a spe-cific system. For several systems the operators A and S may be constructed by simple inspection. To facilitate this task we express the creation and annihilation operators in terms of two Hermitian Operators a and /?, namely ri] = ^{a-03) (3.25) and V = -^{a + iP) (3-26) so then the operators A and S are given by A = -i[a,p] (3.27) and S = a2+P2 (3.28) which allows one to write the operator P as P = q0 - iqa[a, /3] + qs(a2 +/32) (3.29) 8 The usefulness of this formulation is that it makes the form of the operators more self evident when studying specific systems. One can solve for the eigenstates of a system by defining the lower bound state | 0 > or the upper bound state | N > of a system and then solving the equation for 77 | 0 > for the lower bound state or 77+ | N >, whichever is appropriate. 9 Chapter 4 The Quantum Harmonic Oscillator Consider the Hamiltonian for the Quantum Harmonic Oscillator (QHO) given by = | 1 + \mu2x2 (4.1) Define two operators a and P so that a = wx y/fy (4.2) and P = (4-3) Construct now the Antisymmetric operator A and the Symmetric operator S from Q and /?, namely A = -i[a,0] = -i^[x,p] = ^hu (4.4) and 5 = a 2 + f = \muj2x2 + (4.5) 2 2m By inspection we see that the eigenvalues are 10 sk = Ek (4.6) and ak = \hijj (4.7) We see that Ek > \hu (4.8) and Ek+i -EK = huj (4.9) Iterating this result and using EQ = ^fko as the lower bound energy we arrive at Ek =E0 + khu> = (k + ^)hu (4.10) where k = 0,1,2,3, . . . . In this system no upper bound exists. Using a and /? we can construct the creation operator ry* and annihilation operator r/ J = ± ( « - i f » = (4-11) V=±(a + iP)= (4-12) A coefficient Ck is defined by \Ck\2=(k+l) (4.13) The creation and annihilations operators for the Q H O can now be formulated in a compact manner, 7 7 * = ^ Cn | n + 1 >< n | = ] T y/[{n + l ) ] | n + l > < n | (4.14) 11 and the annihilation operator will have the form T] = Yl Cn-i \ n-l><n\=^2 V[n] | n - 1 >< n | (4.15) n n As one can see then rj+ Tj = | C„_ i | 2 I n >< n | = n | n >< n | (4-16) Let us define the energy of a state | n > so that H = en \n><n\ (4.17) where Een is the energy eigenvalue for state | n >. For the Q H O , En = nh w which means that expressed in terms of the creation operator and the annihilation operator 77, the energy of a state | n > of the Quantum Harmonic Oscillator is H I n > = {n+\) hu I n > = TJ + ^) fiw | n > (4-18) 12 Chapter 5 Solving for the | 0 > state for the Quantum Harmonic Oscillator Let us now solve for the lower bound state | 0 > for the Quantum Harmonic Oscillator using the Algebraic Method first proposed by P . A . M . Dirac. [2] Define the lower bound state | 0 > by the condition r\ | 0 > = 0. This means that V \ 0 > = ^ ( x + ^) | 0 > = 0 (5.1) Reformulate this defining equation we arrive at . d mux. , „ - mux. , „ „ ( - + — ) | 0 > = ( D + _ ) | 0 > = 0 (5.2) where D is the differential Operator. The solution to this differential equa-tion is i ^ „ , —mux2. ._ „. \0>=C exp ( 2 f i ) (5.3) where C is a normalization constant which is found to be C = ^ . ( 5 . 4 , so that the lower bound state | 0 > for the harmonic oscillator is given by 13 .mui.i —mux'. .„ „. Having found the lower bound state | 0 > the next state is arrived at by application of the creation operator 7j* to the lower bound state, namely | 1 > = J?f | 0 > (5.6) From the expression for 77* we see that | n > and | n — 1 > are related by I n >= - 7 - I n - 1 > (5.7) V n In terms of the lower bound state | 0 > one can see by inspection that any other state of the Q H O is given by However by inspection we see that n - l II Cm = ^n\ (5.9) m = 0 so then the state | n > expressed in terms of the lower bound state | 0 > is given by ' U > = ~Jn\ (T?t)" 1 ° > ( 5 J 0 ) This description of the Quantum Harmonic Oscillator (QHO) using creation and annihilation operators (also know as raising and lowering operators) is well known. 14 Chapter 6 Angular Momentum Consider the commutation relationship given by [3] J3 = -i[Ji,J2] (6.1) Let Q = Ji P = Ji (6.2) so then q0 = qs = Q and qa = — 1 then A = -i[a,0\=i[Ji,J2] (6.3) and S = a2 + 0* = J\ + J\ = J 2 - J 2 (6.4) Introduce the eigenstates | A , \x > of the operators J 2 and J\ so that J 2 | A , / x >= A | A , / J > (6.5) and J\ | A , / i > = / i | A , / i > (6.6) 15 Consider the raising and lowering operators that act on the eigenvalue fi for a given A. By inspection we see that ak = -Mfc (6-7) and sk = X-nl (6.8) We see then that A > /z2. and that A > fikit^k ± 1) and that A - fi2k+1 - A + /z2. = -Mfc+i - fJ-k (6.9) which means that Hk+i = + 1- Iterating this result we get fik = Ho + k where k = 0,1,2, From the remaining consistency equations we see that A = M O ( ^ O - 1) A = HN(VN ~ 1) (6.10) It is evident that / J Q and /x^ are related by tiN=Vo + N (6.11) where A* is the integral number of steps needed to go from the lower bound state to the upper bound state. Let N = 2j. Solving for A A = j(j + 1) (6.12) where j is an integer or a half-integer. We see therefore that no = —j and [xN = j as expected. It is straighforward to formulate the creation and annihilation operators rj* and 7] in terms of the Euler angles 6 and <f>, namely , t = + ih) = - - J + = - - e * ( _ + i c o t * ^ ) (6.13) and ^ ( J , - ^ , . ^ . . * . - ^ (6.14) 16 The coefficients Ck axe given by Ck =< k + 1 177* | k >= - - ^ < j,n | J+ | / i > (6.15) so that I | 2 = + 1) - MM + 1)] = \ii +1* + 1)(J - (6-16) Let us solve for the upper bound state | A, A > for this system. Define state 7?t | A, A > by 77+ | A , A > = 0 (6.17) This means that Use separation of variables and define | A, A >= 0A,A(#) e" i A* (6.19) Reformulate this defining equation we arrive at h e«x+1» ( ^ - Acotfl) 0 A , A (0) = 0 (6.20) the solution to this equation being 0 A , A (0) = (sin0) A (6.21) so then the upper bound state becomes | A, A >= (sin#)A eiA<* (6.22) 17 The next state down becomes | A, A - 1 >= C(A, A - 1) 77 | A, A > (6.23) where C(A, A — l)is a normalization constant. Any other state then becomes | A,/ j >= C(A, ft) I J * " " | A, A > (6.24) which in terms of Legendre Polynomials becomes | A , , * > = ( - l ) ' * [ ^ | A ^ } ] * ^ ( c o 8 t f ) e ^ , (6.25) 18 Chapter 7 The Rigid Rotator Consider the Hamiltonian for the three-dimensional rigid rotator for the radial wave function u = rR(r), [3] H=— SL + llJL JfL (7 1) 2m dr2 2mr2 2mr2 where we have expanded the term 1(1 + 1) which by inspection allows us to set a and /? in the following fashion a = 7jL <7"2> and It is them evident that the anti-symmetric and symmetric operators are given by J - « [ ! , ! , (7.4) and fc2 j 2 t2/2 and that qa = qs = 1 and go = 0. 19 The eigenvalues of A and S are given by at = (7-6) 8, = E , - ^ l (7.7) where / = m a 2 is the moment of inertia of the rotator. The consistency relations means that in2 Ei >2at = -f (7.8) and E,+i - ai+i - Ei+ai = o ( + 1 + at (7.9) or E l + 1 - E , = 2a,+1 = j{l + l) (7.10) from which we iterate to get Ei = E0 + y £ = E0 + g 1(1 + 1) (7.11) k=l Now since £ 0 = 0 the eigenvalues of the energy are EI = YJ L(L + 1) ( 7 - 1 2 ) No upper bound exists for this system. The creation and annihilation operators and 7? are given by and 20 ft J d . ^ T ^ r - r f ( 7 " 1 4 ) The coefficient Ci are given by I C, |2= l-Et = jjl{l + 1) (7.15) So then with I = m r 2 the moment of inertia of the rigid rotator, C< = 2 ^ ( ' + l ) (7-16) The creation and annihilations operators for the rigid rotator can be formu-lated in a compact manner, r,t = £ C l \ l + l x l \ = ' £ ^jWl(l+l)]\l + l><l\ (7.17) and the annihilation operator will have the form V = J2 C,-1\l-l><l\=J2 ^/] Wl(l - 1)] | / - 1 X l | (7.18) 21 Chapter 8 Solving for the Lower Bound | / > state of the Rigid Rotator Define the lower bound state | Z > by the condition r? | / > = 0. This means that ly/m r ar Reformulate this defining equation we arrive at ( Z - r 4-) | * > = 0 (8.2) ar The solution to this differential equation is \1>=Q rl (8.3) where C; is some normalization constant. Having found the lower bound state | / > the next state is arrived at by application of the creation operator to the lower bound state, namely |1 + 1 > = ^ | Z > (8.4) In terms of the lower bound state j I > one can see by inspection that any other state of the rigid rotator is given by 22 \l + m> = ™> \l> (8.5) l l n = 0 ° ' + " that is where / = m r 2 is the moment of inertia of the rigid rotator. 23 Chapter 9 Spherical Symmetric Potentials The Algebraic Method is a straightforward technique to characterize spherically symmetric potentials. [4] To begin, introduce the radial operators for f f = (x2+y2 + z2)* (9.1) and for p P= - Px + -PV + -Pz (9.2) r r r The operators satisfy the usual commutation relation [ f,p ] — ih. The momentum is given by — d ih .„ „, p = -ih- (9.3) or r Consider a Hamiltonian with a spherically symmetric potential V(r), H=±(j>l+Pl+pl) + V(r) (9.4) We know from straightforward algebra that r2(pt+pl+pl)=rV + P (9.5) 24 so then the Hamiltonian is given by * = ^ 2 + £) + vir) ( 9 - 6 ) which, in terms of normalized eigenstates | nl > of the principal quantum number n and the total angular momentum L, becomes Consider the operator C\ that takes eigenstate | nl > onto | n'l + 1 > where n labels the energy. Then C,* Ci = 2mHi + Fi (9.8) and Ci C* = 2mH[ + Gi (9.9) where F/ and Gi are yet to be determined scalars. We see then that Ci C*i Ci | nl >= {2mEJl + Ft) Ct\nl> (9.10) with EJ1 the energy of the eigenstate | nl > and that C, C, CI | nl >= (2mH,+i + Gt) Ct\nl> (9.11) Solving for Hi+i we get C« \nl>=[ E? + ( F | G ' ) ] C i | n l > (9.12) which means that | n/ > is an eigenstate of with eigenvalue E? = E? + { F l ~ ^ l ) (9-13) If Fi = Gi the energy is constant and nl — n. In this case | nl > and Ci \nl > are states with the same energy and we have a degenerate system. 25 By inspection we see that Ci | nl >= A " | n'l + 1 > (9.14) Similarly, we can show that C* \ n'l + 1 > is an eigenstate of Hi, that is C ; \n'l + l >= fif | nl > (9.15) Note that A" and / j " are complex conjugates of each other as can be seen from \f = < n'l + 1 | Q\nl >=< nl \ C\ \n'l + l >*= tf* (9.16) From these equations we can derive the following simple relationship, | A," | 2 = 2mEp + Fi = 2mE?+l + Gt (9.17) which yields the recursion relationship I A" | 2 - | \?U | 2 = Ft - G,-! (9.18) If this series terminates at some stage with Cj | nl >= 0 then the energy is given by E? = (9.19) It is necessary to determine whether the series terminates on a case by case basis . Consider operators Ct linear in momentum p, namely Ci=p + f(r) (9.20) Since we know that it is simple to see that f{r),p] = ihfr (9.21) C;C, =p2- P(f + f*) + ih^ + ff (9.22) 26 and similarly C,C* = p2 + p(f + / • ) + ih^ + ff (9.23) Studying the Hamiltonian Hi given above we see that there are no terms linear in p so then we have the subsidiary condition / = - / * (9-24) In terms of our Hamiltonian we see then that these two equations for C f C ; and CiC* yield -ih%- -f2 = 1(1 + l)% + 2mV + F, (9.25) ar r J and i h ^ - f = (l + \)(i + 2)% + 2mV + Gt (9.26) ar r^ The equations are easily solved to give f ( r ) = i h l ± ± + ± ( F l - G l ) r + A (9.27) r 2/i and 2mV(r) = (Ft - G , ) 2 ^ - ^ ( F « - Gt)r +(l + l)(Fi - Gi) - \(Fi + Gi) - A 2 - 2ih(l + 1 ) - (9.28) I r where A is an imaginary constant of integration. The potential has to be independent of I, so the coefficients of r, r2 and r _ l must be independent of /. This restricts the possible values of (Fi — Gi) and A in the following way: from the term in r _ 1 we that A is either 0 or proportional to (/ + 1), and from the term in r2 that (Fi — GI) is a constant. The coefficient of r, which contain the product of A and (Fi — Gi) must now be zero showing that one or both of A and (Fi — Gi) is zero. We shall examine these three cases. 27 Case 1: Free Field Set A = 0 and (Ft - Gi) = 0 then and f(r) = ih1— (9.29) y ( r ) = _ _ L ( F ; _ G ( ) = - A ( 9 . 3 0 ) This means that the operator Ci is given by Ci=p + ih(l+l)- (9.31) r Since F[ is independent of r, and V of / they must be constant. The Hamiltonian for the free field case is given by and the energy by EJ1 =< nl | i f , | nZ > J- [<nl\ p2 \nl>+ <nl\ ^ i i l i l | n / > ] _ A (9.33) 2m r1 2m The first two terms are greater than or equal to zero and cannot be simul-taneously zero. Thus apart from the state / = 0 and p2 \ nl >= 0, the energy is greater than V} = — ^ . For the special case when I = 0 and p2 | nl >= 0, we have only one state as Ci I nO >= 0 with energy Ef1 = We see that the series \ ™ , \ " + 1 , \ ™ + 2 , . . . does not terminate and we have infinitely degenerate levels. 28 The energy of the free field case then is El = ^ [| A," I2 -F, ] (9.34) but since we cannot find an expression for A " this means we have a contin-uous distribution of energy levels. Case 2: Coulomb Potential Field Set Fi = Gi and A ^ 0 then the potential is given by 2mV(r) = -Fi - A2 - 2ih(l + 1)- (9.35) r Since we want the potential to be real and independent of I we must have A = ~ w h <9-36> and F ' = D + J ^ ( 9 - 3 ? ) where B and D are themselves real. The potential then has the form V(r) = (9-38) Notice that the first term is nothing more than a constant and so we are free to set it equal to zero, giving us V(r) = ~ (9.39) mr and and the operator C ; in this case is given by 1 I ' R C l = p + ih(l + l ) - - j ^ (9.41) 29 Since Fi = Gi the operator Ci leaves the energy unchanged and we have the recursion relationship | A n , 2 _ | A n | 2 _ B2 B2 _ B2 (21 + 1) which has the solution \ K \ = j ^ [ l + K(l + l)}1/2 (9.43) where K is some real constant. This means that the energy is given by Ef = — (9.44) The Hamiltonian for this system is 2m r1 mr By inspection we see that if B is negative (a repulsive potential) the energy will be positive and if B is positive (an attractive potential) we can have either positive or negative energy. If the energy is positive then K is positive, which means that A " can never be zero and we have infinite degeneracy. If the energy is negative the square root in A™ will vanish at K(lmax + 1) = — 1 which sets an upper bound to /. If we let (Z m a x + 1) = n then we have the familiar result B2 ' 2m n 2 v ' with the operator Ci given by Ci \nl>= ——- [(n + l + lKn-l-l)}1'2 \ nl + 1 > (9.47) n(l + 1) where I = 0 ,1 ,2 , . . . , ( n - l ) . So we see then that for a repulsive potential we have an infinite number of degenerate, positive energy levels. For an attractive potential we have either a finite number of degenerate negative energy levels, or we have an infinite num-ber of degenerate positive energy levels. 30 Case 3: The Isotropic Harmonic Oscillator Let ,4 = 0 but Fi ^ G, ^ 0. Letting V(r) be independent of I we have # 2 2 "<*•> = ™? <9-48> With Ft = B (21 + 3) (9.49) and Gt=B (21 + 1) (9.50) which on the surface appears to lead to an operator Ci of the form - .h(l + l) iBr , n c i , Ct=p + i— - + -r- (9-51) r n where B is a real constant. However, since V(r) is a quadratic potential in B we must be careful to take this into account. We have in fact two inequal operators Ci and Dt given by , .Ml + l) iD2r C,=p + i^ L — (9.52) r n and - M +1) iD2r A = P + t— - + —r- (9-53) r n with Ci | nl >= AJ1 | n'l + 1 > (9.54) and Dt = \nl >= | nl + 1 > (9.55) Since Ft ^ Gi, n ^ n ' ^  n". 31 Consider the equations that result from the C[ operator, namely Ep+i = ~ ~ (9-56) unless Ci \ nl >= 0, which may continue onto 2 D 2 E?+1 = E ? - — (9.57) unless Ci+i \ n I + 1 >= 0, and so on. The energy is decreasing, but it can never be negative as all terms in the Hamiltonian have positive eigenvalues. This means that the series must terminate at some point with E ? = -§n- = { n + l ) D 2 ( 9 " 5 8 ) This means then that = [(«-!) + \)D2 (9.59) which means that n = n — 1. It is easy to show that C, | nl >= Xf | n - 11 + 1 > (9.60) using the recursive relation (with A£ = 0) I A,n | 2 - | A ^ 1 | 2 = -4B2 (9.61) To within a phase factor A? = B[ 2(n - 1) ] 1 / 2 (9.62) Studying the degeneracy of the system the allowed states are | nn >,| nn — 2 >, | nn — 4 > . . . so that 1 can only take the values n, n — 2, n — 4 , . . . to 0 or to 1. Wi th regards to the operator £>i we see that Di | nl >= (j? | n + 11 + 1 > (9.63) however we can show that this series does not terminate, which means we cannot perform useful calculations with the £>, operator. 32 Chapter 10 A Charged Particle in a Magnetic Field: Landau Levels As an example of the Algebraic Method applied to a real discrete system, con-sider an electron in a magnetic field with a symmetric gauge given by A=-*(-y,x,0) (10.1) with the Hamiltonian given by H = ^ + t o ) + ^ - K ) + 2 ^ (1°-2) The eigenvalue solution to this problem is known as the Landau Levels of charged particles in a magnetic field, and was solved in a complete and detailed fashion in 1930 by Dr. L . D. Landau. [5, 6] We begin by defining for the conjugate momenta the following eB nx - Px + zc eB *v = Py ~ 2^ KZ=PZ= hkz (10.3) The equations of motion for this system are then given by 33 d f » i h [ " v , H ] mc ~ where | * , = ^ r „ * ] = 0 (10.4) eB to = — (10.5) mc The solution to these equations of motion are TTX = TTlLj(y0 - y) TTy = mu>(x — Xo) ftx — Pz = constant (10.6) The Hamiltonian can be rewritten as * = + * + £ f < * 2 + ^ + s s < * - « * • > = + S ( 1 0 7 ) Let us define two operators n± — Trx±iny as step up and step down operators such that HT = T— (7T+7T- + 7T-T+) = 7T+7T_ + \ fo>J (10.8) 4m 2m 2 where for state | <fin > | <})n >= CnTT+ | (fin-l > (10.9) with Cn being a normalization constant for the state. It is straightforward to show that Cn = (2nmftw)i (10.10) 34 Define the ground state of the system such that 7T_ | <po >= 0 7T2 | <f>o >= hk\<j>0> (10.11) Then it is straightforward to show I 4>n > = , , „ n \ , , < U o > (10.12) v/(2nn!m/TXj) T To find the state | 4>o > w e a P P ' y the explicit form of T T _ and 7r 2 on state o > so then 7T_ | <j>0 >= -ihlfc + ~ ' ^° > irz = -ih— I 4>0 > (10.13) oz so then the state | 4>o > has the form I <Po >= }{x ~ iy) exp [ikz] exp [- ^^J^] (10.14) where f(x — iy) is a yet to be determined function which reflects the infinite degeneracy of the system and A = ( — )i (10.15) mu If we impose an additional constraint on | <f>o >, namely | <j>o > should be an eigenfunction of XQ then f(x-iy) = A exp [ - 2 - ( x - i y - 2a)2} (10.16) where X Q | <t>o >— o. \ <j>o > and x0 \ 4>n >= a \ <j>n >. 35 We therefore have a state \ </>o',a > given by | (f>0; a >= Cn(exp [ikz] exp [ - ^ [ ( * 2 + V2) + (x-iy- 2a)2}}) (10.17) The normalization constant Cn is given by C„ = ( ^ e x P [ ^ ] (10.18) yielding 1 i a 2 1 \4>o\a>= ( ^ j ) * ( e x P ^ ] e x p [ - ^ [ ( x 2 + 2 / 2 ) + ( x - i y - 2 a ) 2 ] ] ) (10. To obtain the higher energy eigenfunctions | <f>n',a > we use the step up operator TT+ to arrive at 1 > = ( 72 ) n ( A r ^ ) < e x P ^ e x P i - ^ x 2 + y ^ +\x-iy-2a)2} })HN(^) (10.20) with the energy of level n given by £ ( „ : p , ) = M » + i ) + | ; which describe the Landau Levels of an electron in a magnetic field. 36 Chapter 11 Second Quantization The Algebraic Method can also be used to characterize Second Quantization of Boson and Fermion fields. [2, 3] Define a number operator N = 77+77 ( 1 1 . 1 ) so that the number operator N is N = \{a2+f32)+l-[a,(}} ( 1 1 . 2 ) We will denote the eigenvalue for the number operator N as vn. To focus on the character of bosons and fermions we shall study them separately. Bosons : For bosons we have the commutator [77,77+] = 1 (11.3) which means for bosons A = -i[a,0] = l (11.4) and S = a2+/32 =2N + A ( 1 1 . 5 ) so then qs = —qa = \ and go = 0 . 37 Applying the consistency relations we get an = 1 (11.6) and sn = 2vn + l (11.7) It follows then that v„ > 0 and vn+i — Vn — 1- By iteration we see that vn = vo + n (11-8) with n = 0, 1, 2, . . . . It can immediately be seen that the lower bound is UQ = 0, and that no upper bound exists. The coefficient | Ck | 2 is given by \C„ \2=^(sn + an) = vn + l = n + l (11.9) As we can see then for bosons the creation and annihilations operators can also be formulated in a compact manner. To within a phase factor, the creation operator will have the form rjt = ] T y/(n + l) | n + 1 >< n | (11.10) n and the annihilation operator will have the form 77 = ^ sjn I n - l x n I (11-11) n Fermions : For fermions we have the commutator 7 7 ^ - 7 ^ 7 7 = 1 (11-12) which for fermions means A =-i[a,p] = S-2N (11.13) and 38 S = a2 + p2 = 1 Applying the consistency relations we get (11.14) = 1 - 2u, n (11.15) and = 1 (11.16) Applying the consistency relations we get two conditions 1 + (1 - 2vn) > 0 (11.17) and 1 - (1 - 2vn) > 0 (11.18) which means that 0 < vn > 1. As well 2 — 2(vn+i + vn) = 0 which means This in turn means that v — 0 and vn = 1. Since the.spectrum is dis-crete, this shows that the only eigenvalues for the fermions number operator is vn = 0 o r l . These two discrete eigenvalues alternate as the order number n grows as is seen in the expression un+i = 1 — un. The coefficient | Ck | 2 is given by vn+\ — 1 — vn (11.19) I C n | 2 = x(Sn + O N ) = 1 - J/ n = V„+i (11.20) 39 Part II Similarity Transformations 40 Chapter 12 Introduction To describe a quantum system means choosing a Hilbert space of states on which the canonical variables are defined as operators. In turn, this means that definite representations of the canonical (or anti-) commutation relations has been chosen. For the case of quantum systems with a denumerably finite number of de-grees of freedom, all representations are unitary equivalent to each other. This fact may be used to our advantage when attempting to solve the Schrodinger equation for the quantum system. The idea behind the Unitary Similarity Transformation method is to begin with a known eigenfunction and transform from a Hamiltonian with a simple form and structure to the more involved Hamiltonian and then use the trans-formation to find the transformed eigenfunction. In the second part of this thesis I look at several examples of Unitary Sim-ilarity Transformations and how they can be used to simplify Hamiltonians describing quantum systems. Examples of the Similarity Transformation Method include a method to determine the ground state eigenfunction using a generating function, Electron-Spin Resonance, the Foldy and Wouthuysen Transformation and an approach first proposed by Wentzel and applied by Schwinger to describe the non-relativistic interaction of an electron with a field. Schwinger used this approach to solve for the Lamb shift of the electron in a central coulombic potential. 41 Chapter 13 The Similarity Transformation Start with the Schrodinger equation (h = 1) i ^ = H ^ (13.1) Consider the unitary transformation given by [7] V>' = U ip (13.2) Inverting this transformation we have «/> = [/+ V>' (13.3) which in terms of the Schrodinger equation in Hamitonian form id-^f=HUU' (13.4) If the unitary transformation does not explicitly depend on time then from which we find = U H C7+ ip' = ip' (13.6) 42 where H* = U H W. If the transformation depends explicitly on time we find .dW tp' .dW . . r r t d%l>' at at at so then djp_ at H U* il>' - i^- M> at which in turn yields (13.7) (13.8) = •[ U HU^-iU^-]xl! (13.9) 43 Chapter 14 The Fundamental Theorem of Algebra Applied to the QHO The Fundamental Theorem of Algebra applied to second order differential equa-tions states that an equation of the form F = a D2 +bD + c (14.1) where D = ^ is the differential operator, can always be expressed in terms of the product of two linear expressions in D, namely F = a D2 +b D+ c = a0 {D-ri){D-r2) (14.2) where r\ and r% are the roots of the second order differential equation F. Returning to the example of the Quantum Harmonic Oscillator, and taking note that p = - p t _ ihjL (143) ax the Hamiltonian can be reformulated in the following fashion, H = a0(p- a) f ( p - a) = a 0 (p f - a f ) (p - a) (14-4) where ao = 57^  and a = imcox. 44 This relationship is expected since <n\H | n >= E <n\n> (14.5) which sets the constraints < n | (p - a)1 = (p - a) | n > (14.6) When expanded out the Hamiltonian for the Q H O becomes H = a0(pt p — pt a — o) p + a)a) (14.7) however we know that p^ p — p2 and that a* = —a so then this equation becomes H = a0(p2 + a + a + p a + a p) (14.8) Using the commutation relation for the a term [a,p] = i h ^ (14.9) we get H = a0(jr + a]a + 2p a + ih^) = a0(p2 + a1a - ih^-) (14.10) Ox Ox We see then that H = ao(p — a)* (p — a) = oo(p2 + 0*0 + hmu) (14.11) So then 1 0 mu)2x2 huj .„ . „„. ^ = 2 ^ P + — + T ( 1 4 J 2 ) where a rescaling of H can be done to remove the zero point energy Ezp= ^ (14.13) 45 Chapter 15 A Similarity Transformation Applied to the QHO Consider now the term (p — a) to be an operator acting to the right on a state | u > namely (p-a) \u>=(-ih-4- ~a) I u >= -ih(4~ + %) \u> (15.1) ax ax in Introduce a unitary transformation operator eF such that eF e'F = 1 (15.2) which we introduce between the operator and the state that it is operating on - " < ; § + i " > = - < + ^ i - > <15-3> where | w >= e~F \ u >. Expanding out the expression we see -ih(— + —) eF \w>= -ih(eF — + eF D - — eF) \ w> dx in ax h = -iheF (DF + D-l4) \w> (15.4) n 46 What we have is an expression of the form (p - a) | u >= eF(p-a + pF)\w> (15.5) We are free to choose the function F to meet our needs. Returning to the Quantum Harmonic Oscillator, let p F = a that is DF = ^ where a = imcox, we get dF ia dx h The solution of this equation is (15.6) F We see then that /—muj x , —muj x2 .„ „ — a - d x = ~ 2 h - ( 1 5 - 7 ) 17 , . muj x2 . , „ _ „. w >= e~t | u >= exp [ 2 ^ ] | w > (15.8) So what we have found is (p - i m u i ) | u >= exp [ — — — ] (p) exp [ 2 f e ] | u > (15.9) It is now straightforward to find the state | 0 > by solving the equation P ( [ - ^ P ^ ^ ] | 0 » = 0 (15.10) which has the solution [ e x p Z l r ~ ] | 0 > = c ( 1 5 - n ) where C is a constant of integration, yielding (to within an arbitrary phase factor) _, , —mux2 . , „ . | 0 >= C exp [ — ^ — ] (15.12) In solving the equation for the state | 0 > we have used an operator to transform the lowest energy state, that is 2 | 0 >=» [ exp m " . x } | 0 >= | w > 0 (15.13) In 47 Let us look again at the Hamiltonian H <u \ H | u >=< u | a0(p - a) + (p - a) \ u> (15.14) Applying the similarity transformation (p-a) | u >= eFpe'F \u> (15.15) Consider now the adjoint, namely ( eF p e~F | u >)t = (| u >) + ( e - F ) t (p) f (eF)+ =< u \ eF (p)t e - F (15.16) This means the transformed Hamiltonian equation becomes < u | (p — a)* (p — a) | u > = < u | e f (p)* e~F eF p e~F \ u > = < w | (p)* p | «; > (15.17) That is 1 + < u I i f I w >=' — < w I (p)T j6 I it; > (15.18) 2m Let us confirm this by straightforward calculation. For the Q H O we know that state | n > in the number representation is given bv \n>=Cn Hn[(y/—)x ] exp ^ (15.19) where Hn is the Hermite polynomial and Cn is the normalization constant for state | n >. This means that the transformed state | w > is given by | w >= e~F | n > mux2 , _ „ r . ,mw. . —mux2 = exp - 7 — [ C„ i f „ [ ( > / - r - ) a ; ] exp 2ft " " l v v ft ' J " 2ft (15.20) 48 Applying the momentum operator p to the state | w > yields - i . „ ,hmu> „ ,, .raw. . p \w>=-i 2n Co Hn-! [(y/-jr)x ] (15-21) The Hamiltonian then is ^ - <w \ (p)i-F eFp | «; >= - L | eFp | w > | 2 (15.22) im 2m In terms of the representation given above 1 i F ~ i 1 9 — \e p \ W > \ 1 r _ 2nhmu) 2 f°° mu 2 - m w i 2 (15.23) as expected. 49 Chapter 16 Formal Expression for the Similarity Transformation Having shown its application to the Quantum Harmonic Oscillator let us now express the similarity transformation operator in a more formal fashion. Consider the Hamiltonian equation given by <u\ H0 | u >= <u \ p \ u> (16.1) 2m Let us try to express a different Hamiltonian H = Ho + Hi by transforming the Hamiltonian Ho, using a unitary transformation eF such that < u | eF e~F Ho eF e~F \ u >=> < w | Ho+Hi \w> (16.2) where the state | w > is given by | w >= e~F | u > (16.3) and where e~F Ho eF = Ho + #1 (16.4) From the transformation we get [8] e~F H0eF= H0+ [H0,F] + ±[[H0,F},F} + ^ [[[H0,F],F},F] + ... (16.5) 50 So then we must find F such that e~FH0eF = H0+ [H0,F] +^[[H0,F],F) + ...= H0 + H1 (16.6) or solving for Hi H1= [Ho,F} + ^[{Ho,F},F} + ^[[[H0,F],F],F} + . . . (16.7) The function F is made up of fundamental dynamical variables. It can be constructed from generalized coordinates and their conjugates or from creation and annihilation operators. Let us return to the example of the Quantum Harmonic Oscillator with the Hamiltonian given by where Ho = j ^ p 2 . We shall construct the function F in terms of the posi-tion operator q. Let us try the Ansatz F = aq2, so that ^ q 2 = [HQ,aq2] + ^[[H0,aq2],aq2]+ ... (16.9) Evaluating this expression term by term, [9] Wo, aq2}= f - \p2, q2} = ^ (pq + qp) (16.10) 2m m It is worth noting the symmetrization of this term. The next term is l [ [ i f o , a q2],a q2] = ^ [(pq + qp), a q2} (16.11) It can be seen that this term is equal to ±[[H0,aq2],aq2} = ^ q2 (16.12) A l l further terms in the expansion being zero since they commute with F = aq2. 51 This means then that the eigenvalue for the | w > is given by | w >= exp (-aq2) \ u > (16.13) where we see that ( m W ) ^ 2 = 2 f c V q 2 ( 1 6 1 4 ) 2m m so then a = — (16.15) We have recovered the expression for the Quantum Harmonic Oscillator us-lg a similarity transformation of the free field hamiltonian HQ. 52 Chapter 17 The Generating Function In terms of powers of the position operator q it can be shown that any term of the form an q" will lead to an expression of the form A i = £ [ n(n'i)an q n ~ 2 + n 2 a ' q 2 n ~ 2 ] { i 7 - i ] If you have a function / which is a polynomial in q then f(q) provides the following expression Consider then the function F made up of a sum of powers in q, namely oo F = J2 anqn (17.3) n=0 We have then the following '±± = Y , n ( n - \ ) a n qn~2 (17.4) and d1 n=2 (f)2 = [ E na„ qn~l } [ £ mam qm~' ) (17.5) a q n = l m = l so then we arrive at 53 2m 1 dq2 Kdq' J £ [ ( 2 a 2 + a 2 ) g ° 4- (6a 3 + 4a x a 2 ) q + (12a 4 + 6 a i a 3 + 4a 2) g 2 + (20a 5 + 8o ia 4 + 12a 2a 3) q3 + (30o6 + 10aiO5 + 16a 2 a 4 + 9a 3) g 4 + (42a 7 + 12aio 6 + 20a 2 a 5 + 24a 3 a 4 ) g 5 + (56a 8 + 14aia 7 + 24a 2 a 6 + 30a 3 a 5 + 16a2) g 6 + (72a 9 -I- 16aia 8 + 280207 + 36a 3 a 6 + 40a 4 a 5 ) q7 + (90a 1 0 + 18aia 9 + 32a 2 a 8 + 42a 3 a 7 + 48a 4 a 6 + 2f>a\) g 8 + ( l l O a n -f 20aiaio + 36a2a9 + 48a 3ag + 56a 4 a 7 + 600506) g 9 + (132a i 2 + 22aian + 40a 2 a i 0 + 54a 3 a 9 -I- 64a 4 a 8 + 70a 5 a 7 + 36a2.) g 1 0 + . . .] By inspection we see that the lead term in each power of qn has a coefficient given by (n + 2 ) ( n + l ) (17.6) each mixed term anam has a coefficient given by 2nm (17.7) and each repeated term anan has the coefficient n 2 (17.8) The mixed indices in each line start from the right with a i a m where m +1 = n + 2 and continue up to a n _ 2 an_2 for even powers, of n , or a n _ 3 a n _ 2 for odd powers of n. 54 Chapter 18 The Hermite-Lindemmann Transcendental Theorem Having derived an expression for a type of similarity transformation it is worth noticing the limitations set out by the Hermite-Lindemmann Transcendental Theorem which states that a function of the form G(q) = An exp[F„(g)] (18.1) n cannot be equal to zero. This means that the function F(q) outlined above could not apply for state eigenfunctions that have a zero in the domain, that is, an eigenfunction that intercept or crosses the x-axis. To accommodate eigenfunctions with a zero in the domain we need to con-sider functions of the form D(q) = Y P»Qn e x P £ q U ( 1 8 - 2 ) n m or equivalently D{q) = exp [In £ j3nqn + ]T an qn] (18.3) n m It is a well known theorem that for a symmetric potential that the eigenfunc-tion of lowest energy has no nodes. Only symmetric potentials will be considered in this thesis. [10] 55 Chapter 19 The Base State of the Hamiltonian Before proceeding to study some example Hamiltonians it is worth noting the expression we get when we set each term of the function F(q) in powers of qn equal to zero, namely (2a 2 + a 2 ) = (6a 3 + 4aia 2 ) = (12a 4 + 6 0 ^ 3 + 4a 2) = . . . = 0 (19.1) Expressed in terms of a i this yields the following exponential function Q = exp [ aiq - ^a2q2 + i a 3 g 3 - ^a\qA + ...] = e x p [ E ( - i r + 1 ^ ^ ] (19.2) n = l The series in the exponential is the familiar Taylor series expansion of l n ( l + a\q) that is In (1 + axq) = axq - ^(ax q)2 + i ( d g) 3 - ^ (o i g) 4 + . . . = JT ^ 2 (19.3) n = l provided | ax q |< 1 and a\ q ^ —1. We find then that fi = exp[ f] ( - 1 ) " + 1 {^-^~ ) = exp[ ln ( l + alQ) ] (19.4) 56 Let us now consider fi to be an eigenstate of the Hamiltonian H. We see that H0 | n > = ^-p1 p | fi > = 0 (19.5) Zm Consider now the momentum operator acting on fi, namely p | fi >= -ih4- | fi >= -ih— I fi > (19.6) dq (1 + a x q) 1 v ' We can make the expression real by setting a\ = i, in which case ^ > = ( T T ^ ) l n > ( 1 9 7> (or we can set a\ = 0 which leads to a trivial solution). Let us set a\ = i, to give a function | fi >- exp[ l n ( l + iq)] (19.8) As one can see then for the momentum operator p < fi | p | fi >= ?i (19.9) If we need we can redefine the momentum operator p so that p=p-h= - f i ( i + t i L ) (19.10) in which case < fi | p | fi >= 0, while we still retain [q,p] = [q,p} = ih (19.11) 57 With the rescaling of the momentum operator p p we have a rescaling of the Hamiltonian namely h2 H0=*H0 + — (19.13) 2m For the original Hamiltonian Ho we have < ft | H0 I n >= o (19.14) and for the position operator q we have < ft | q I ft >= 0 (19.15) We can express | ft > in the following equivalent fashion \Q>=p exp(iO) = | ft > n o r m (19.16) where and 6 = arctan(g) (19.18) In this fashion, we have defined a normalized base state | ft > for the system. 58 Chapter 20 Example Hamiltonians Let us now build up two example Hamiltonians using the generating function: The Quantum Harmonic Oscillator The expression ^ [ n(n - 1.) aqn~2 + n2a2nq2n-2 ] (20.1) provides a clue as to how best to build up the expression for any Hamiltonian H. Let us take as an example the Quantum Harmonic Oscillator where #! = ^mu>2 q2 (20.2) If n = 2 we get two terms, one in q° and the second in q2, so then a.<i / 0 in the series for F . We know for the Q H O the zero point contribution is a term of order q°, namely HZp = Y (20-3) so by inspection we see that £ l + = £ ( 2 0 - 4 ) 59 or solving for a 2 «2 = - 2 a 2 + -2fi ( 2 0 - 5 ) where we recognize the familiar term The first term — \ a\ can be considered the base state contribution. Continuing on and solving for a 3 we see and for a 4 and for 05 03 = \a ? - \ (T ) a i (20-6) 1 4 1 ,mw. o 1 ,mu).o ,™~s a 4 = - 4 a " + 3 ( - ^ ) a ? - l 2 ( - ^ > ( 2 ° - 7 ) a s = 5 a ? - 3 h r ) a ? + 1 5 ( x > a i ( 2 0 - 8 ) and for a 5 1 fi 1 , m u . 4 17 .mu.o 2 1 ,mu ; .o ° 5 = "6 a ? + 3 ( ! T ) a > - 90 ^ a ? + 45 ( - f t - } ( 2 0 ' 9 ) and so on to higher order. Grouping the terms we have F = a i q - \ a \ a 2 + \ a \ a * - \ a \ q * + \ a \ q5 - i a? q* + ... mai 9 1 , m u . o r 1 , m u , 2 1 , 4 + [ J ( T ! W - B < X > , « J + 5 < T ! ' , > * , + - < » 1 « » We recognize the first set of terms as being In (1 + a\q). 60 Rearranging terms we get = In(1 + a i q) + — ^ 1 - - - - _ £ _ + _ tftfXJMUl) We see then that in terms of the transformation exp(F) = exp [ ln ( l + alQ) + ^ q2 [ 1 - 2 G l 9 2/i * 1 3 (1+ 01 9) nee 2h or + ) ]] (20.12) exp(F) = exp [ln(l + aiq)] exp [— <?2 [ 1 -2h * 1 3 (1+ 01 9) m U } a, 2\ 11 + -2H » { q ) ]] (20.13) We can therefore express the lowest energy state of the Quantum Harmonic Oscillator as w > = e x p ( - F ) | fi > = e x p [ - l n ( l + aiq) ] e x p [ - ^ q2[ 1 - \ ° ' Q in 6 (1 + a,\ q) fi > (20.14) We are free to set the value of a\ = 0 so then we have which, apart from a normalization term, is what we expect to find for the Quantum Harmonic Oscillator. What has been done is starting with the base state | fi > we have used a similarity transformation to transform the base state into the ground state of the Quantum Harmonic Oscillator. 61 Returning to the transformation of the Hamiltonian HQ we see e~F H0eF = ^ - e " V e~Fp eF (20.15) 2m Expanding out each operator we have for the annihilation operator e~Fp e F =p + \p,F]=p-[F,p]=p-ih^-F (20.16) aq and for the creation operator e " V EF =tf+ \p\F]=tf -[F,pt] = -p + ih4-F (20.17) aq which for the case of the Quantum Harmonic Oscillator yields (oi = 0) -F - F - ... ^ „• „ l t m w e~tpet = p-ih—F = p-ih— q (20.18) aq ft and e~ p' e = —j3 + ih—F = —p + ih—— q (20.19) do ft We can now go on to build up the other states of the Quantum Harmonic Oscillator in the usual fashion. 62 The Hi = A q Hamiltonian Let us take as a second example the symmetric potential given by Hi = X | q | (20.20) We see then for this potential we start with a i and find for 0,2 for 03 for (14 for 05 for 06 a2 = -\a\ (20.21) a 3 = r ' + i ( ^ ) ( 2 0 - 2 2 ) a 4 = _ I a 4 _ I ( ^ ) a i (20.23) a 8 = l a ? + ^ ) a ? ( 2 ° - 2 4 ) 1 fi l . m A . o 1 , m A , , .„„ a a = - 6 ° ? - 6 ( l F ^ - 3 0 ( - ^ ) . ( 2 ° - 2 5 ) and so on to higher order. Grouping the terms we have F = a i 9 - \ a2 q2 + i al q3 - ± a\ q4 + ^ a\ q5 - ± a° q6 + ... l . m A , o l . m A , 4 l . m A , 2 s l . m A , ~ fi or F = I n ( l + 0 l 9 ) + i ( ^ ) 9 3 (1 | o i 9 + q2 - | o ? g 3 + . . . ) 63 By inspection we see that the next term is of the order Repeating the same procedure as outlined above for the Q H O we have for the state I w > . l.m\. o 1 , m A . , fi . . 4 , m A . 4 7 . . W >= C n o r m e x p [ - - ( - ^ ) q* - _ ( _ ) » f - O W W ? q ) + - ] (20.29) where CnOTm is a normalization constant. Expanding out the momentum operators we see e Fp eF = p — ih-^-F aq * P ~ ^ ) ? 2 + \ ( ^ ) 2 <? + ^ ( | ( ^ ) 4 * 6)] (20-30) and e - " ^ e F = -p + ih-^-F aq *~P + ihfg) <72 + f + *(fj|(^ )4 a6)] (20.31) We can now go on to build up the other states of the system in the usual fashion. 64 The H\ = a q2 + /? q4 Hamiltonian Let us take as a third example the potential given by Hi = a q2 + ft q4. We see then for this potential we start with ai and find for a2 a2 = ~ a \ (20.32) for 03 a-3 = (20.33) for <i4 ^ = ( 2 0 - 3 4 ) for a 5 1 r 1 , m a , .„„ „„, «5 = - i g ( - ^ - ) a i (20.35) for a6 for a7 1 fi 1 .met. , 1 , m S , , a « = - 6 o ! + 1 5 ( - ^ ) ^ + 1 5 ( # ( 2 0 ' 3 6 ) 1 7 1 , m a . o 2 ,m/?, „„, O 7 = 7 o I - 1 5 ( ^ - K - 1 0 5 ( ^ ) 0 1 ( 2 ° - 3 7 ) and so on to higher order. Grouping the terms we have F = o i 9 - ^ a2 q2 + | a 3 g 3 - J a} g 4 + | of g 5 - i a? g 6 + i a\ q7 + ... 1, m a . x 1 , m a , * 1 . m a , , fi 1 , m a , , 7 1 -J71/9, fi 2 ,m/3. 7 + 1 5 ( ^ } 9 " 1 0 5 ( ^ ) a i q + (20.38) 65 or T-T . /•, \ l.ma. 4 1 , m a , e 1 , m a . 2 fi 1 , m a . , 7 = ' n ( 1 + a i *> + 6 ( ^ } 9 - 1 5 ( ^ ) a i 9 + 15 ( q - 1 5 ( ^ ) a ? 9 1 ,m/?. fi 2 ,m/J , 7 We can simplify this equation to read (20.39) F = ln(l + a i q ) + \{™^) g 4 ( l - g + | a 2 a 2 + . . . ) + » K ^ ) ( ^ ) «»] (20.40) Wi th I w >= e x p ( - F ) | ft > we set ax = 0, to find . l . m a . 4 1 , m ^ . 6 . . . m a m l 8 l . I - > = e x p [ - - ( ^ ) g 4 - - ( - ^ - ) * » - * [ ( _ ) ( _ ) g«] (20.41) Let us find the momentum operators for this example. e Fp eF = p — ih-^-F aq ,^d,l,ma. 4 l.mB. fi nr,ma.,m8. 8 l = P - ^ 6 ( ^ } 9 + I S ^ 9 + 9 ] = P ~ iK[\C^) , 3 + \Cf) <? + * [ 8 ( ^ ) ( ^ ) 9 7 ) (20-42) and e e F = — p + i f t - j -F ag „ .. rf.l.ma 4 1 m/3 6 ma mP 8 = - P + t ^ - ^ ) 5 3 + 5 ^ ) <75 + * f ) ^1 (20-43) We can now build up the other states of this system in the usual fashion. 66 Chapter 21 Describing Electron-Spin Resonance Using A Similarity Transformation Consider a two level spin system of an electron in a magnetic field with a Hamil-tonian given by [8] H = ^ [ B0 az + Bi(<7+ exp(- iwi) + cr_ exp(iwr)) ] (21.1) The unperturbed energy eigenvalues are given by E± = ± ^ (21.2) where u0 = ~yB0, with B0 being the unperturbed magnetic field and B\ be-ing the perturbing field. To solve for the eigenstate of the Hamiltonian we shall use a similarity trans-formation given by I V ( i ) > = exp(-iut^) I x(t) > (21.3) The transformed Schrodinger equation becomes i9 ^ > = ^ [ (w0 - w) az + 7 Bi(c+ exp{-iut) +<r_ exp(twt)) ] I > (21-4) 67 where we have transformed the a to a±(t) — exp(iu;£^-) a± exp(—iut^) (21-5) where a± = \{oi ± ia2)- In the case of the a± it is easy to show that a±(t) = cr ± exp(±iw<) (21.6) so then the transformed Hamiltonian becomes i~^gf~~ = \ I (wo - <7j + 7 exp(-iwt) +<r_ exp(io;i)) ] | x(i) > (21.7) It is worth noting that az and a± are in the Schrodinger picture. The solution to the transformed Schrodinger equation is | X(t) >= exp[ ( i n | ) ( c o s ( ^ ) + sin(0a 2)) ] | </>(0) > (21.8) where fi cos(#) — (UIQ — w) •fi sin 0 = 7.81 (21.9) from which we get the explicit expression | tp(t) >= exp (ioj^-) [cos(fi^) - i sm(U^)(cos(6az) + sin(6>CT2))] 11/>(0) >= U(t,0) \ ip{0)> (21.10) From this solution we can extract the Resonance behaviour of the two level spin system. 68 Chapter 22 Describing the Interaction of an Electron with a Field We shall now use a Similarity Transformation to describe the interaction of an electron with a electromagnetic field. [11]. The Schrodinger equation for a non-relativistic particle interacting with a field is ihd^it) = [-L(p+ e-A)2 + V(r) + Hrad]*(t) (22.1) Zm0 c where dt$(t) = 9 . Upon making the Similarity Transformation *'(*) = exp (22.2) A(t) = exp [i^HL] A exp [ J l ^ ] the new state vector satisfies the equation ihdt*'(t) = H' *'(*) (22.3) where the transformed Hamiltonian is given by 69 Now perform a second Similarity Transformation to remove the "virtual effects" induced by the second term (dropping the primes): *(*) exp [-tS(t)]*(t) (22.5) in such a fashion as to define S(t) using the equality at m0c Let the vector A(t) be derived from a vector potential Z(t), namely A(t) = (22.7) then, since the momentum operator is a time-independent operator, S(t) = — (p • Z(t)) (22.8) m0c Looking on the transformation operator [8, 28] , dexp[-iS(t)] .8S l t „ OS, , n n n . e x p l S { t ) ^ = - . ^ + -2[S, w ] (22.9) exp iS(t) | ? exp [-iS(t)] = ^ + [S, ^ ] (22.10) with the series terminating because [A, A] is a c-number. The transformed Schrodinger equation is m m = l £ n Z + 2 [ S ' d t ] + e X p [ i S { t ) ] V { r ) e x P [ - i 5 ^ + 2 m ^ e x P [iS(t)] A(t)2 exp [-iS(t)} ]9(t) (22.11) 70 Using the expressions for S(t) and its time dependence outlined above the transformed Schrodinger equation becomes TP1 ih, p m q , ( t ) = [2m~0+ jij—ZlP-ZW^-Mm+expliSit)} V{r) exp[-*<?(*)] ]*(*) p2 i ih, e 2 _ T7/_ i eZ, ^ 2 T O o + 2 W b , - Z ( ' ) ' p - i 4 W 1 + V ( r + ^ ) 1 * W (22. where the A2 has been dropped. In the dipole approximation, with A(t) = YekfcV-^— [ akn exp -iukt + a*kflexpiukt ] kp z ( f ) = ' 5 Z € f c M c v / 2 ~ 3 [ aknexP[-iukt] - a*ktlexp[iujkt] } kn * (22.13) the commutator of A(t) and Z(t) (with a suitable upper limit to the inte-gration) becomes [Ai,Zm\ = — 6im dw - T 3 7o I" 2 ' (22.14) So then [AhZm} = - l ^ c 5 i m (22.15) In terms of a = — , the commutator is 2 ftm0c m 0 2 m 0 with 2e 2 8e 2 71 The kinetic energy of the electron will then be given by " ' - ( I = £ (22-18) 2mo mo 2m where the electron mass has been renormalized with m = mo + dm being the observed mass as opposed to mo the bare electron mass. Note that the effects of the radiative corrections are included in the trans-formed expression for V( r ) namely expiS(t) V{f) e x p - i S m = Vlf + — Z) (22.19) mc Upon expanding out the transformed expression for V(r) the Schrodinger equation becomes ihdt9{t) = V(f) + — Z • W ( r ) + i ( — ) 2 (Z • V ) 2 V ( r ) + . . . ] *(«) 2m mc 2 mc (22.20) Note that there is no virtual interaction for a free electron because Z-VV(f) is proportional to the force acting on the electron and therefore to its acceler-ation. For a free electron all radiative corrections have been incorporated into the mass renormalization. If no photons are present initially or finally the Schrodinger equation becomes %hdt9(t) = [£- + V(f) + \{ — ? < (Z • V ) 2 >vac V(f) + ...) *(*) (22.21) 2m 2 mc since in this case < Z >vac= 0. The remaining leading vacuum expectation term in the Schrodinger equation can now be evaluated to yield < (^Zf >vac= ( £ |4 = - ( - f ^ ^ (22.22) mc mc 2wjJ n mc OJQ 72 so the Schrodinger equation becomes ihdt$(t) = + V(f) + — ( — ) 2 In ^ - V 2 V(r) ] (22.23) 2m 7r mc OJO For the case r V 2 V(f ) = 47rZe2o(r) (22.24) the perturbing term results in the familiar Lamb shift which is given by 73 Chapter 23 The Foldy-Wouthuysen Similarity Transformation One of the more interesting uses of a Similarity Transformation was proposed by L . L . Foldy and S. A . Wouthuysen to decouple the Dirac equation into two two-component equations, where one of the two-component equations reduces to the Pauli description in the relativistic limit and the other describes the negative-energy states. [12] Following the procedure outlined by Foldy and Wouthuysen we shall use a series of transformations to remove from the Dirac Hamiltonian all operators that couple the large components of the eigenfunction to the small components. By convention the operators that do not couple large and small components are known as "even" operators and those that do are known as "odd" operators. For instance, 1, /3,S are "even" operators and 0,7,75 are considered "odd" op-erators. The F W Transformation Applied to A Free Dirac Particle Consider the Dirac equation for a free particle given by the Dirac Hamilto-nian H = a-p + /3m (23.1) Consider then a unitary similarity transformation U — exp(iS) with the operator S hermitian and not explicitly time-dependent, then ip' = exp(i5) ip (23.2) and (h = 1 and c = 1) 74 = exp(iS) H ip = exp(iS) H exp( - iS ) ip' = H ip' (23.3) at such that the Hamiltonian H is to contain no odd operators by construction. Try the Ansatz exp(iS) = exp( /3 a • p 0(p) ) (23.4) so then the transformed Hamiltonian H becomes H = a - p ( c o s [ 2 | p | 0 ] - — sin [ 2 I p I » ] ) I P I +P(m cos [ 2 I p I 6}+ I p I sin [ 2 I p I 0]) (23.5) In order to eliminate the "odd" operator (a • p) set tan[ 2 I p I 6) = i ^ i (23.6) m which means the transformed Hamiltonian J / ' is = ^ [ m 2 + p 2 ] (23.7) which has eigenvalues which can easily be found using conventional methods. 75 Chapter 24 The FW Transformation Applied to an Electron in a Field Consider now the Dirac Hamiltonian for a charged particle [12] H = a-(p-eA) + pm + e<j> (24.1) Note that P a • (p - e A) = -a • (p - e A) p (24.2) and 0 e (j) = e (p p. It is worth remembering that the Hamiltonian may be time dependent. In this case the operator S will also be time dependent and so then it is not possible to construct an operator S that will remove all operators from the transformed hamiltonian H . We are, however, able to expand out the hamiltonian in a power series in ^ keeping terms to what ever order we wish in this non-relativistic expansion. Introduce the time dependent transformation ip' = exp(iS) ip (24.3) 76 so then i— exp[—iS]ip = H ip = H exp( - iS ) ip dib d - i = exp(- tS) i-^ + iigi e*P[-iS\)il> (24.4) which yields i-tp' = [ exp{iS](H - i — ) exp[-tS] ] V ' = & (24.5) Expanding out the expression we see Q [ exp[iS]{H--i^)exp[-iS] = H + i[S,H}- i [ S , [5, ff]] - ^ [ 5 , [5, [5, £]]] + [5, [5, £]]]] + . . . (24.6) We see that the first order term is given by 8m + e <t> + tf + i[S,8]m (24.7) where we have let d = a • (p — e A). To remove the odd term = a • (p — e A) we choose S = The remaining terms become i[S,H] = -0+JL[0,e4>] + -Bd2 (24.8) and ^ 5 J 5 J 5 ^ ] ] ] = ^ - ^ [ . , [ . , e ^ ] - ^ 3 (24.9) and (24.10) 77 and l[S[S,[S,[S,H}]}] = ^ (24.11) while the next few terms in 4^ are dt ~ 2m dt (24.12) and By inspection it is possible to see that the odd terms now appear only to order m To reduce the odd terms further it is necessary to apply a second Foldy-Wouthuysen Transformation of the form §' =-i£[£l*,e + (24.14) 2m L 2ro l ' Y 1 3 m 2 2m dt* K ' Under this second Transformation we see that H" = [ exp[iS](H - i—)exp[-*5'] = = /3m + e <j> +1?" (24.15) where •d" is now of order . Applying a third transformation of the form ST = ^ d" (24.16) 2m results in the Hamiltonian 78 It is now possible to explicitly evaluate the terms, and 2m 2m 2m K ' so that [a-p,a-E] = 8 m 2 6 16 6 V • £ + —— 6- V x E + —rb -Exp (24.20) 8 m 2 8 m 2 4 m 2 By inspection we have One is then free to interpret the individual terms of this transformed hamil-tonian in the usual fashion. 79 Part III The Bogoliubov IVansformation 80 Chapter 25 Introduction The Bogoliubov Transformation is used for Diagonalizing a Quadratic Bosonic Hamiltonian. Several examples of Bogoliubov Transformations are presented, including phonons in a system of weakly interacting particles, as well as Electron Spin-Resonance. The Bogoliubov Transformations will also be used in simple models of cos-mological particle creation presented in a subsequent section of this thesis. 81 Chapter 26 The Bogoliubov Transformation Let us consider a system of bosons with a quadratic Hamiltonian H with off diagonal elements in the creation operator and annihilation operator n. The commutation relations between H and will be of the form = I>i Aik+ViBik) (26.1) i The adjoint of this expression is [H, Vk] = ~ £ (Vi Aik + 4 Bik) (26.2) i where for simplicity we shall consider only real A and B . We would like to find a transformation that will diagonalize the Hamiltonian. [13] We can do this by introducing a new set of bosonic creation and annihilation operators a* and a so that <*t = L folu*" - ^ ( 2 6 - 3 ) k and ^ = £ (m ukv - nl vk„) (26.4) k which fulfill the following commutation relations 82 [ at, <*l ) = 0 (26.5) [av, a„]= 0 (26.6) >„, al ) = 6VII (26.7) and [H, al } = Ev at (26.8) What we see then is [H, at] = £ ( [H,V$\ U*x ~ \&M = E» £ fo* * V - ^ * it (26.9) Rewriting this relationship we see that k,i £ [ Tii[(Aik - 6ikEv) Ukv + BikVkv ] + n { BikUkv + {Aik + 6ikEv)Vkv ] = 0 (26.10) This means that we obtain the following simultaneous equations £ [ (Aik - 8ikEv) Ukv + BikVkv } = 0 (26.11) * and £ [ BikUkv + (Aik + 8ikEv)Vkv ) = 0 (26.12) k which must hold for all values of i . We can express these equations in matrix form, so that AU + BV = UE (26.13) 83 and BU + AV = -VE (26.14) where the matrix E is a diagonal matrix. It follows then that {A + B)(U + V) = AU + BV + BU + AV = (U - V)E (26.15) Multiplying this expression on the left by (A-B) yields (A - B)(A + B)(U + V) = (A - B)(U - V)E = (AU + BV + BU- BV)E = (U + V)E2 (26.16) Define a matrix W such that W = (A-B)~1/2 {U + V) (26.17) so that (U + V) = (A - BY'2 W (26.18) We then see that {A-Bf2 {A + B){A-BY/2W = W E2 (26.19) or succinctly MW = W E2 (26.20) where M — (A - B)1'2 (A + B) (A- Bfl2 (26.21) Since A and B are real matrices they must be symmetric in order that H shall be hermitian. This in turn means that M is real and symmetric and the problem of diagonalizing the matrix M is a standard one. Having solved the expression MW = WE2 for W we can generate (U + V) and (U — V) and in this way find transformation coefficients such that the ex-pression [H, al } — Ev aj, can be satisfied. 84 Having satisfied this expression it is now necessary to satisfy the three com-mutation relations noted above. These three equations impose a rigid set of restrictions on the transformation coefficients U and V , namely [ at, at ] = Y ( U ^ ~ VkliUkv) = 0 (26.22) k which in matrix notation is UTV - VTU = 0 (26.23) and [ a ^ a l ] = Y,(Uk»Uk" - VkliVkv) = (26.24) k which in matrix notation is UTU - VTV = 0 (26.25) When diagonalizing the matrix M choose the normalization of the eigenvec-tor in such a fashion that WTW = E~l (26.26) This means that WT (A - B)1'2 (A + B) (A- Bfl2 W = E (26.27) Since A and B are symmetric matrices it follows that [{A - BY'2]T = (A - B)1'2 (26.28) so then WT {A-B)l/2 = (U + V)T (26.29) This means that (.4 + B){A- B)1'2 W = (U + V)E (26.30) 85 so we see that (U + V)T (U — V) = 1. The transpose of this expression can be written as WT (A - B)1'2 {A + B) =E(U- V)T (26.31) This means that {U - V)T {U + V) = 1. What this means is that provided we normalize the eigenvectors according to WTW = E~l then the new creation and annihilation operators cfi and a will automatically fulfill the boson commutation relations. An Example Using the Bogoliubov Transformation Consider phonons in a system of weakly interacting particles, described by the Hamiltonian [14]. H ~ £ ek a\ak + \ Yl V(kl ~ fci) °*i °*2 a *2tafcit A(fci + k2 - kv - k2i) k 1 (26.32) where the delta assures conservation of wavenumber. Let N0 = a ja 0 , and assuming Vk = V-k then the Hamiltonian can be written as H = £ ek a\ak + ±(N0)2 V0 + N0V0 ^ o [ o t k k +JV0 £ Vk a\ ak + i y V 0 £ Vk( ak a_* + a\ aik) + higherterms (26.33) k k where the summations do not include k =0. Reading from left to right the terms in the Hamiltonian are the following: (a) Kinetic energy: £ €k a\ak (26.34) k (b) Interactions in the ground state: aj aj <z0 ao (26.35) (c) One particle not excited in the ground state: 86 a0 a'k ak oo (d) Exchange of one particle in the ground state: (26.36) ak a0 ak a0 (26.37) and a0 a'k aQ ak (26.38) (e) Both initial and final particle in the ground state: % a0 ak a-k (26.39) and a'k a_k a0 ao (26.40) Terms with three ground state operators are excluded by momentum con-servation. We can take the expectation value of 7V0 + 53 a \ ak t o D e the number of particles N in the system. Collecting terms we have a reduced Hamiltonian in bilinear form Express this reduced Hamiltonian in terms of new boson operators a* and \(N0)2 V0 + Y^+NVk)al o f c + i j V ^ V f c ( o t a_* + alka\) + ... (26.41) a such that [H,ak] = Xal (26.42) [H,ak] = - A ak (26.43) and [ak,a[] = 6kk' (26.44) 87 The first two expressions are satisfied if the new Hamiltonian is written in diagonal form, namely Hnew = £ Xk a\ Q f c (26.45) k In terms of the reduced Hamiltonian Hred we have Hred - \N2 VO = £ (26.46) where Hk = w 0 (a\ ak +a}_k a-k) + u>i(ak a-k + al f c a| . ) (26.47) with wo and ui\ given by w 0 = ek + NVk (26.48) and « i = NVk (26.49) Make the transformations ak = ukak - vko)_k (26.50) and a\ = uka\ - vka-k (26.51) where uk and vk are real functions of k. For these transformations we see that [ak,a[] = ul-vl (26.52) which means we should choose uk and vk to make the commutation relation equal to 1. We can do this by letting ak =uk a k + vk a^_k (26.53) 88 We see then that [H,a[] = uk(cj0 a[ + w i a_*) +vk(ujo a_ f c + wi a[ = X(uk a\ - vk a-k) (26.54) This means then that u>o uk+ ui vk = A uk (26.55) and wi uk + OJ0 vk = - A vk (26.56) These equations have a solution provided A 2 = w g - uj = (ek + NVk)2 - (NVk)2 (26.57) The normalization of the commutation relation is assured if we choose uk = cosh(xfc) (26.58) and uk = sinh(x/c) (26.59) This means that the Hamiltonian H is diagonal if tanh(2 X f c) = (26.60) ek + I\Vk The ground state in terms of the transformed annihilation operator ak has the property ak | $ 0 >= 0 (26.61) 89 We see that akak = u\ a\ak + v\ + v\ a^_ka-k + uk vk( ak a[_k + ct-k ak) (26.62) which means that the mixtures of excitation k in the ground state | $o > is given by < $o I a\ak | $ 0 >= v2k = ^( cosh(2xfc) - 1) (26.63) where cosh(2X*) = (ek + NVk)[(ek + NVkf - N2V2]-1'2 (26.64) 90 Chapter 27 Field Quantization and Spin-Resonance Let us now quantize the radiation field and take into account the reaction of the spin with the field. Consider spin 1/2 particles in a magnetic field with the main field in the z-direction. The energy of the field is given by [15] H field = hue) a (27.1) The interaction energy is given by Him = fi^oy + M a f + a,)ax (27.2) with K a coupling constant. Let ax = a+ + <r_ then H = H fieid + Hint = hujo)a + hu>o^Y +hK(a^ + a) ( C T + + a-) (27.3) This Hamiltonian can be approximated in a simpler form as H = H fieid + H^t = hua^a + ftuo-^ +hK(aj a-+ a a+) (27'.4) 91 This approximation is equivalent to decomposing the linearly polarized cav-ity R F field into two opposite circularly polarized waves and keeping only the one rotating in the same direction as the spin precession. This simplified Hamil-tonian is still hermitian. Let us split the Hamiltonian into two new operators H = fi(Ci + C 2 ) (27.5) where C\ = u(a'a + C2 = K{S+ + S - ) - ^ < J Z AUJ = w — wo S+ = a+a S- = a-a} (27.6) It can be shown as well that C\ and C2 commute with H and with each other. When there is no coupling between the particle spin and the radiation field we have a complete set of basis states consisting of the states | n > for the radi-ation field (with a)a | n >= n \ n >, and the states | ± 1 > for the spins where a, | ± 1 > = ± | ± 1 > . Using the usual procedure we can use this complete set of states to describe the state vectors of the coupled system. We immediately find that for C\ Ci | n , ± l >= w ( a f a + y ) | n, ±1 >= w(n ± ^) | n, ±1 > (27.7) and so C\ is diagonal in this representation. It is easy to see that C 2 is not diagonal in this representation, but since C\ and C\ commute we can use a Bogoliubov Transformation, which is a special form of similarity transformation, to find a representation in which they are both diagonal by taking linear combinations of the eigenkets of C\. In this new representation, the transformed Hamiltonian is diagonal so that we can easily solve the Schrodinger equation for the system. 92 Consider the new state vectors given by | ip(n, 1) >= cos# n | n + 1, - 1 > + sin#„ n , +1 > | tp(n,2) >= - sin 0„ | n + 1,-1 > + cos6n | n, +1 > (27.8) where | n + 1, — 1 > corresponds t o n + 1 quanta in the field with spin-down and | n, +1 > corresponds to n quanta with spin-up. The ground state is given by | 0, — 1 > with no quanta and spin-down. This state is considered separately. The angle 6n is a parameter where n may take on any value between 0 and oo. We see that the new states are diagonal and normalized, and are orthogonal to the ground state. We see as well that We see that the two states | </?(n, 1) > and | y(n,2) > are degenerate eigen-states of C\ irrespective of a choice in 8n. Applying C 2 to the new eigenstates we obtain Aw C 2 | <p(n,l) >= (Ky/(n + 1) s i n 0 n + — cos0„) | n + 1, —1 > Ci ¥ > ( n , l ) > = w ( n + - ) \<p(n,l)> (p(n,2) >= w(n+ i ) | <p{n,2) > Ci | 0 , - 1 >= - i u | 0 , - 1 > (27.9) + (ny/(n + 1) cos^„ - — sin^„) | n , + l > where the composite operators S+ and 5_ were used, namely S+ | n + 1,-1 >= y/(n + 1) | n , + l > S+ | n , + l >= 0 S_ | n + 1,-1 >= 0 5_ | n,+l >= V(n + 1) | n + l , - l > (27.11) 93 In order for | <p(n, 1) > and | <p(n, 2) > to be eigenvectors of C2 we must be able to choose 6„ so that C 2 | <p{n, 1) >= A n | <p(n, 1) > C2 \ <p(n,2) >= \'n |y>(n,2)> (27.12) By inspection we see that if we let t a n * n = ? ^ ± i } (27.13) | A w + A„ and *n = Vl(^f)2 + K2(n+l)} (27.14) then | <p(n,l) > and | <p(n,2) > are eigenkets of C2 with eigenvalues ±A„, and where A n = — A n . For the ground state we have C2 | 0 , - 1 > = ^ | 0, —1 > (27.15) which we see is also an eigenket of C2. The eigenvalues of the transformed Hamiltonian then are H | v>(n , l )>=f i [u (n+^) + A„] \<p(n,l)> H \<p(n,2) >= h[u(n + ^) - Xn] \<p(n,2)> H | 0 , - 1 > = - ^ | 0 , - 1 > (27.16) It is worth noticing that the eigenstates of the transformed Hamiltonian are mixtures of eigenstates of the unperturbed Hamiltonian Ho-lt is possible to express the eigenstates | n, ± 1 > in terms of | ip(n, 1) > and | <p(n, 2) > where (n = 0 ,1 ,2 , . . . , oo), namely n + 1,-1 >= cos0 n | tp(n, 1) > - s m 6 n \ (p(n,2) > | n , + l >= sin0„ | <p(n, 1) > + cos0„ \ ip(n,2) > (27.17) 94 Consider now the transition probability between and initial state | n, + 1 > ( a state with n quanta and spin-up) and a final state | n + 1, — l > ( a state with n+1 quanta and spin-down). We see that |< 7i+ 1,-1 I exp( — ) I n , +1 > | 2 = sin 2 26nsin2 Xnt n - (A^+'LU'+i) ™% ^ <a"'!+4«2<-+»i <27-18> It is worth noting that even if n = 0 there is a probability the spin will flip and emit a quanta of light which may later interact with the system. The model outlined above therefore takes into account spontaneous emission of quanta. 95 Part IV The Dressed Oscillator 96 Chapter 28 Introduction It is possible to describe the coupling between an electromagnetic field and an oscillator in terms of a dressed oscillator. I shall consider a non-relativistic sys-tem of oscillators coupled linearly to a scalar field in ordinary Euclidean 3-space. I start with an analysis of a non-relativistic system of oscillators confined in a reflecting sphere of radius R, and assume that the free space solution to the radiating oscillator is obtained by taking the radius of the cavity arbitrarily large in the R-dependent quantities. The limit of an arbitrarily large radius is taken as a description of the radiating oscillator in free space. From a physical point of view we give a nonperturbative treatment to the oscillator radiation introducing some coordinates that allow to divide the cou-pled system into two parts, the "dressed" oscillator and the field, what makes unecessary to work directly with the concepts of "bare" oscillator, field and in-teraction to study the radiation process. 97 Chapter 29 An Exact Approach to Oscillator Radiation using A Contact Transformations Consider a harmonic oscillator qo(t) with natural frequency wo coupled linearly to a scalar field (f>(x, t), the whole system confined to a cavity of radius R centred on the oscillator. [16] The equations of motion are r R q0(t) + ulq0{t) = 2-ngc \ d3r 4>{x,t)8(r) (29.1) Jo | ^ - V2cf>(r,t) = 27rgc q0(t) 8(r) (29.2) where g is a coupling constant. Using Spherical Bessel functions in the domain 0 < r < R can be rewritten as a set of equations coupling the oscillator to the harmonic field modes, namely D oo q0(t) + u20q0(t) =vYl (29-3) t=0 qi(t)+u2qi{t) = nujiqo(t) (29.4) where n = y/2gAw and Aw = TTC/R is the interval between two adjacent field modes, A w = w , + i — w, = TTC/R in the spherical cavity. 98 Let us consider how a harmonic oscillator couples to N other oscillators. In the limit N —>• oo we recover our original situation of the coupling oscillator field after an appropriate redefinition of diveregent quantities, in a manner similar to renormalization in field theory. In terms of the cut-off N , the coupled equations are rewritten taking N as the upper limit instead of oc for any summation and the system of N + 1 coupled oscillators qo and qi is represented by the Hamiltonian, 1 N H = + Wo?o ] + £ P 2 + W J 9 J 1 - 27?wi<?o<Zj (29.5) This Hamiltonian can be turned to principal axis by means of the canonical transformation (29.6) performed by an orthonormal matrix T = T£, n = (0, k),k = 1,2, . . . i V . The subscript 0 and k refer to the oscillator and the harmonic modes of the field respectively. Let r refer to the normal modes, r = 0, ...N, then the transformed Hamilto-nian in the prinicpal axis is H = lJT(P? + nlQl) (29.7) where the Qr are the normal frequencies corresponding to the possible col-lective oscillations modes of the coupled system. Using the coordinate transformation gM = T^Q„ in the equations of motions and making use of the normalization N £ ( T ; ) 2 = 1 (29.8) /i=0 we get the following conditions on the orthonormal matrix T (29.9) 99 and N TS = [1 + Y v2"2 K 2 - fi2)2 -1/2 (29.1G) and " l - n l = v 2 £ - ^ ( 2 9 - n ) j=l U3 There are N +1 solutions fiM to these equations, corresponding to the JV +1 normal collective oscillation modes. Take = fi so that we get - iVr, 2 - n 2 = V2Jt (29.12) It is easily seen that if wg > Nrf then there are only positive solutions for fi2, which means that the system oscillates harmonically in all its modes. There is also an oscillation mode whose amplitude varies exponentially and that does not allow stationary configurations. We shall disregard this case. We shall assume in our model that LJQ > Nrf and define a renormalized oscillator frequency a; such that LO = - Nn2} (29.13) In terms of the renormalized frequency then j=i UJ We get the transformation matrix elements for the oscillator-field system by taking the limit N -> co, namely T0" fi„ (29.15) and 100 The Eigenfrequency Spectrum Let us return to the coupling oscillator-field by taking the limit N —» oo in the relations outlined above. In this limit it becomes clear why we need the frequency renormalization, which serves as an analogue to mass renormaliza-tion in field theory. The infinite uo is chosen in such a fashion as to make the renormalized frequency Z5 finite. Recall the solutions with respect to the variable Q of the equation * 2 - ^ 2 E ^ (29-17) give the collective modes. Let Uj = = 1,2... ,00 and take the positive x such that Q = then using the Langevin identity ~ a;2 1 z2 = 2 [ 1 ~ 7 r x c o t ( 7 T X ^ ( 2 9 - 1 8 ) 3=12 the equation can be rewritten in the form cot(<,7r) = + — ] (29.19) Kg TTX irgc The secant curve corresponding to the right hand side cuts only once each branch of the cotangent on the left hand side of the equation. Label the solutions xr = r + er, where 0 < er < 1, r = 0 ,1 ,2 , . . . and the collective eigenfrequencies are Clr = (r + er)— (29.20) where the e r satisfy the equation = %f + i6 <2921) The field <j>(r, t) can be expressed in terms of the normal modes. Expanded out the field in terms of spherical Bessel Functions 101 where oo <{>{r,t) = c ]jT qj(t)<t>j(r) (29.22) Using the principal axis transformation matrix together with the equations of motion we obtain an expression for the field in terms of an orthonormal basis associated to the collective normal modes, namely ^ , t ) = c £ g t ( i ) 4 t ( r ) (29.24) k=l where the normal collective Fourier mode is given by $ j r ) = V Tk s i n ( l r 1 U j / c ) 9 k ( r ) Y ' ry/2*R which satisfy the following equation of motion (29.25) O 2 n _ A ) 0 f c ( r ) = 2 7 ^ ] 6(r)Tk (29.26) which has a solution of the form ^ = - ^ 2 T ^ r k ^ T ^ ^ ( 2 9 - 2 7 ) To determine the phase 6k expand out the right hand side of this equation and comparing with the normal collective Fourier expression we see that the phase is set by the boundary condition, namely s i n ( y i ? - < 5 f c ) = 0 (29.28) Recall that flr = (r + e r) ^ it is easy to show that the phase 0 < Sk < n has the form 6k = nek (29.29) 102 Thus solving for the expansion of the field in terms of the normal collective modes yields, 2 ^ | r | V[s in 2 6k + ( ^ g ) 2 ( l - ffe'»)] The infinite R limit In the continuous formalism of free space the field normal modes Fourier components are given by d>a = H(n) ^ du^-^ — (29.31) where the leading term is given by 2gtl V [ ( f i 2 - Zj2) + TTgW} *(«) = n,oi-rt\ • - » ™ (29-32) Splitting u i " n i into partial fractions we get 1 sin(w | r | /c) /oo ^ ^ ( f l i i e ) | r | ./-oo l r l (29.33) where S±(u -Q) = -P{—^TT] ± i J(w - ft) (29.34) where P is the Principal value. The transformation matrix takes on the following form in the limit r -> oo, y/[2g] n^jAQ] AQ^O -ui2) + n2g2n2] Tr - lim V l g J " V t " " J (29 3 ^ and 103 Tr = 2gWjAw Or , 2 g where A u = Aft = ^ . For arbitrarily large R we note that Aw —¥ 0, so then the only non-vanishing matrix elements are those for which OJJ — ftr « A w . To arrive at an expression for the matrix elements in the limit R —> oo let us take R large enough so that A w « Aft and consider the points of the spectrum of eigenfrequencies ft inside and outside of a neighbourhood n of ojj. Note that when R> this means | > Aw, then we may consider R such that the neighbourhood f of Uj contains an integer number n of frequencies ftr, namely n A w = \ = ^ (29.37) If R is arbitrarily large we see that f is arbitrarily small, but n grows at the same rate, which means that the difference Wj — ftr outside the neighbourhood n of w, is arbitrarily larger than Aw,, implying the corresponding matrix element T[ tends to zero. We also see that all frequencies ftr inside the neighbourhood of n of w; are arbitrarily close to Wj, being an arbitrary large number N . Only the matrix elements T[ corresponding to these frequencies ftr inside the neigh-bourhood n of w, are different from zero. For these we make a change of labels r = i - n (ui - | < ftr < Ui) (29.38) r = i + n (w, + - < ftr < Wj) where i — 1,2, Wi th the changed labels we get 7 7 - 1 v / K ^ - w V + ^ W ] ( 2 9 ' 3 9 ) and T ' ± H ~ n ± et y/lW - Jy'+ * v w ? 1 ( 2 9 ' 4 0 ) 104 which satisfies the condition w? - ZJ2 cot(Trei) = (29.41) Using the relationship i °° i 1 ^ r f ( « , ) . - + E[ 5 r i^ + 5 r ^ 5 5 ] (29.42) the normalization condition for the relabelled matrix elements becomes (T/ ) 2 + Y [{Ttn? + ( T / + n ) 2 ] = 1 (29.43) n = l and the orthogonality relation (i ^ k) Y ^ = 0 (29-44) r in the limit R —> oo. The Transformation matrix in the limit g = 0 For arbitrary R it is easily seen that lim9_>o TQ = 1 if QR = ZJ or 0 otherwise. We also see that the matrix element T[ for i ^ r all vanish for g = 0. For small g for the diagonal terms T\ we have - ~ ~ 2£^\ (29-45) 2 C i (n? -ZJ2) (oji + sii) or expanding €j for small g Tt{9 = 0) = 1 (29.46) We see then that in the limit R —> oo the matrix element 7 J remains an orthogonal matrix in the same sense as for finite R. Wi th the choice of the procedure of taking the limit R -¥ oo using a discontinuous formalism from a confined solution, it can be seen that the matrix elements do not tend to the free space limit as it would in the case using a continuous formalism. In the discontinuous formalism all non-vanishing matrix elements T[ are concentrated inside a neighbourhood n of and their spectrum is a quadrat-ically summable and enumerable set. Also, the elements TQ are quadraticaaly 105 integrable expressions. Describing the Radiation Process Using the Dressing Formalism Let us define coordinates q'0 and q't as the coordinates of the dressed oscil-lator and the field. These dressed coordinates will provide a non-perturbative description of the oscillator-field system. The general conditions that the dressed coordinates must satisfy are the fol-lowing: 1) the coordinates q'0 and q\ should be linear functions of the coordinate modes Qr, 2) the coordinates q'0 and q\ should allow the construction of orthogonal con-figurations corresponding to the separation of the system into two parts, the dressed oscillator and the field, and 3) the set should contain the ground state I V The last condition restricts the transformation between coordinates q'^, \i = 0 ,1 ,2 , . . . and the collective ones Qr to those that leave invariant the following quadratic form Y = ^ Ql = W(9 0) 2 + E "Ui? (29-4?) r i Our configuration will behave in a first approximation as completely indepen-dent states, but they will evolve as time progresses, as if there are transitions between states, while the ground state r 0 is a fixed eignestate and does not evolve with time. The new coordinates q'^ describes the dressed configuration of the oscillator and the field quanta. The eigenstates of our system are represented by the normalized eigenfunc-tions 0non i n 2 . . .(Q,t) = H M T QsWoexp-1^^ (29.48) where Hni is the ns Hermite polynomial and Nn, is a normalization coeffi-cient given by Nn, = (2 - " ' n . ! ) - i (29.49) and r 0 is a normalization representation of the ground state 106 8 To describe the radiation process, havings as initial condition that the os-cillator qo be excited, one considers the interaction term in the Hamiltonian written in terms of qo, qi as a perturbation, which induces transitions among the eigenstates of the free Hamiltonian. In this way it is possible to treat the problem approximately having as initial condition only that the "bare" or "un-dressed" oscillator be excited. However, it is well known that this initial condition is physically not consis-tent due to the divergence of the "bare" or "undressed" oscillator frequency if there is an interaction with the field. One traditionally gets around this diffi-culty by a renormalization procedure, introducing by means of a perturbation, order by order corrections to the oscillator frequency. It is possible to use an alternative procedure where we do not make explicit use of an interacting "bare" oscillator and field described by coordinates qo and q, respectively. Instead we introduce "dressed" coordinates q'0 and q'{ for the "dressed" oscillator and field defined by 6 v t ^ f i = £ T l Q t ( 2 9 - 5 1 ) r where UJ^ = ZU, w,-, and is valid for arbitrary R, while leaving invariant the quadratic form outlined above. In terms of the bare coordinates, the dressed coordinates are expressed as ll = £ 1- (29-52) where a<" = -F=- £ T M ZVVO- (29.53) As R becomes a large number we get for the coefficients the following: for Q 0 0 i r°° 2 9 n y n <m k m a 0 0 = - = / — _ 2 * , 2 2 0 2 = A ° ° ^ ( 2 9 - 5 4 fl->oo y/uj Jo (ft — to ) + TT g VI 107 for ai0 lim a,o = lim R—KX Aw->oo y/(jJi (w2 — J2)2 + (7rp)2w! I E n = l 2ti - - } (29.55) for a0i lim aot = lim Aw—>oo (2c/2wf Aw) 5 2 £ i A / W (w? w 2 ) 2 + 7r 2<J 2w 2 ' - - ] (29.56) and for a^-lim a i f c = 8ik (29.57) fl-*oo Thus we can express the dressed coordinates q'^ in terms of the bare ones <7M in the limit R -> oo g 0 = A00(uJ,g)qo (29.58) 9i = 9. It is interesting to compare the finite radius and infinite radius cases. In the case of finite R, the coordinates q'0 and q\ are all dressed, in that they are all collective, both the oscillator and field modes cannot be separated. In the infinite radius limit R -> oo, the coordinates q'0 describes a dressed oscillator, while the dressed harmonic modes of the field, described by the co-ordinates q\ are identical to the bare field modes. In the limit R oo the field retains its bare field modes while the oscillator is accompanied by a cloud of field quanta. Therefore we identify the coordinates q'0 as the coordinates describing the oscillator dressed by the field. The systems is divided into dressed oscillator and field. There is no interaction between them, the interaction being absorbed in the dressing cloud of the oscillator. We can use the dressed coordinates to describe the Radiation Process. 108 The Dressed Oscillator and Radiation Let us define for a fixed instant the complete orthonormal set of functions 4>nQKl...(Q,t) = II W*, 91)] To (29.59) where q'^ = q'0,q'i, and tJ^ = ZJ. This function can be written as linear combinations of the eigenfunctions of the coupled system (let t = 0), namely tfw.fo')= E ^M.:: (° )^on a . . . (Q,o) (29.60) norii... where the coefficients are given by T%z.;:{o) = J' <JQ v K „ K 1 . .>«<,», . . . (29.61) with the integral extending over the entire Q-space. Let us consider the case where there is only one dressed oscillator q'^ in an excited state i>o...oN(,)o...(q') = A ^ i W [ ^ f K ) r ° ( 2 9 - 6 2 ) Using the relationship ^ E W * » [ ^ r r y ] (29.63) T o N °^ (M ) o . . . = (^b) 1 W ° W • • • (29.64) where the subscripts p = 0, i refer respectively to the dressed oscillator and the harmonic modes of the field, with the quantum numbers subject to the con-straint no + n\ + • • • = N. { T o ) m o ( r i ) r o i ^—' m o ! m i ! . . . mo+mi + ...=N from which we get 109 Let us now look at the simple case of N = 1 and look at the behaviour of the " dressed" oscillator q0 in the N-th excited state. In the case of ./V = 1, let Tj be the configuration of the "dressed" oscillator q^ in the first excited state, so that the time evolution of this state is given by r» = Y^F.(t) R M ( 0 ) V /""(*) = £ T ; Tl exp(- tf i . t ) (29.65) 8 We see then that as time progresses the excited "dressed" oscillator shares its energy amongst itself and all other "dressed" oscillators. Starting in its first excited state at time t = 0, the "dressed" oscillator's decay rate may be evaluated from its time evolution operator, namely rS = ^ r w r f ( o ) (29.66) where f00(t) is the probability that the "dressed" oscillator shall be excited and f0,/(t) is the probability that the "dressed" oscillator shall have radiated away a field quanta of frequency uv. Under this formalism we have a radiation process that is a simple exact time evolution of the system. Evaluating f00(t) we have which for large time t and arbitrary coupling constant g, yields | / 0 0 ( t ) | 2 = exp(-ngt) (l + (%?) + exp( -7 r^ ) z^-r (sm(wt) + — c o s M ) ) +(=&Y (29-68) for the oscillator decay probability where U = \J(u2 — (^)2)-In the weak coupling regime where j < w w e find | / 0 0 ( t ) | 2 « exp(-TT^) (29.69) as expected. 110 Chapter 30 Transitions Due to a Forcing Function Let us now consider an oscillator which at time t = — oo is in the ground state. We shall now try to find the probability that at time t = +00 the oscillator will be in the nth excited state if it has been subjected to a force f(t) such that I f(t) |-> 0 as * -> ±00. [17] Consider a Hamiltonian given by which in terms of the creation and annihilation operators can be expressed as We shall try to find a solution to the Schrodinger equation in the form of a compound function namely In differentiating the operator acting on $(—00) with respect to time we must remember that the creation and annihilation operators do not commute, namely (30.1) H = hu(tfa + |) - f(tWj^{tf + a) (30.2) $(<) = c(t)exp[a{t) a*] exp[ a}exp[j(t) a*a] * ( -oo ) (30.3) [ a, a f ] = 1 (30.4) 111 The derivative of with respect to time can be put into the form = G exp[a(t) a)] exp[ a]exp[j(t) a+a] * ( -oo ) (30 where the operator G is given by G = c -— aJa + c( - a - p ) a T dt at at , d/3 „ d7, d c d7 d/3 .„„ Equating the terms in the expression ihG = H w e find d7 da dB dt , + iua = —r—- fit) dt y/(2hfiw) J v ; dt ~ i o j p = 7 (2w m c 2 v dt Given the initial conditions £ - + a { d± _ (30 a( -oo) = 0 8{-oo) = 0 c(-oo) |= 0 ' ' (30 we find the following iexp(—iui) fl .. <. < and for c(t) '• exp(iwt) y/(2hnu) •/—oo 7(t) = —iwt (30 112 c(t) = e x p ( - y ^ ) exp[- J f(t) exp(-iu)t') dt j f(t) exp(zwt") dt } J —OO (30.10) The probability of transition from the state oo) to the nth excited state for t = +00 is Wn = lim I / #*$ dx I2 t-»oo J = lim | c / ** exp[a(<) a f] exp[ a]exp[7(r) a+a] * ( -oo ) | 2 (30.11) i-+oo J If the initial state $ ( — 0 0 ) was a ground state $ 0 then, exp[ a] exp[7(i) a*a] V0 = *o (30.12) since the annihilation operator applied on the ground state is equal to zero. Using the normalized wave function * n - ( a f ) n *o (30.13) then 00 M exp[a(t) a+] * 0 = £ - J - J T * m (30.14) So then we find for the probability of the oscillator being in excited state n having started in ground state 0 is 1 f-+oo n! where Wn0 = lim - t I c(t) | 2 I a(t) | 2 " (30.15) 1 /•+°° I 2 = e x p [ - - ^ — y I y_ /(*) exp(i««) dt \2} = exp(-v) (30.16) 113 and 1 r+°° I a(t) | 2 = ^ 2 h l l u ) I ]_ fit) expiiut) dt |2= v (30.17) We therefore obtain for Wno Wn0 = ^ exp(-i/) (30.18) which is the familiar Poisson distribution. Two Examples of Forcing Functions Let us look at two examples of forcing functions f(t) and solve for v. For a Gaussian type forcing function we have / ( i ) = / 0 e x p [ ~ ] (30.19) which yields a v of while for a simpler type of forcing function / ( * ) = / o 77~~1?1 (30-21) we get As can be seen, the time evolution of a forcing function fit) will result in a finite probability of excitations of higher energy states. Where we may have started in the distant past with only the ground state we have in the distant future higher energy states which are now populated. [30] 114 Part V Cosmological Particle Creation 115 Chapter 31 Introduction In this Part of the thesis I look at uniform acceleration in Rindler space, particle creation in two simple models of a Friedmann-Robertson-Walker Universe, as well as a hypothesis that gravity is an induced quantum effect. 116 Chapter 32 Particle Creation in a Two-Dimensional FRW Universe Consider the two-dimensional Friedmann-Robertson-Walker universe with line element ds2 = dt2 -a2{t)dx2 (32.1) where the space is expanding uniformly as described by the scalar function a2(t). Using the conformal time parameter n defined as n = dt/a(t) we see that the line element is given by ds2 = a2(r]){dr)2 - dx2) = C{rj){dn2 - dx2) (32.2) where C(r?) is a conformal scale factor. As a simple model suppose that the conformal scale factor is given by [18] C{rj) = A + B tanh pn (32.3) where A , B and p are constants. This model represents an asymptotically static universe that undergoes a period of smooth expansion. We see that in the far past and far future we have C{n) -> A±B (32.4) 117 Since C (n) is not a function of x, spatial translation invariance is a spacetime symmetry. This means that we can separate the variables in the scalar function. Consider the scalar field equation given by [ Ux + m2 + £R{x) ](f> = 0 (32.5) where Ux4> = g»v V M V , 1 W ( - f i ) 9»" dv] (32.6) V(-9) with g being the determinant of the metric, and R is the Ricci scalar. Use as an ansatz the function uk (r?, x) for <j> Uk(r),x) = exp(ikx)xk (") (32.7) In terms of Xk (v) w e have d2 ^ Xk(v) + (k2+ C(n)m2)Xk=0 (32.8) The solution of this ordinary differential equation can be given in terms of hypergeometric function. [5] In the remote past we have the asymptotic function which behaves like a positive frequency solution (n -4 -oo) ukn{r],x) -4 -jj-l exp(ikx-icjinT]) (32.9) V ( 4 7 r W i n J where u i n = [k2 + m2(A-B))* "out = [k2 + m2{A + Bp w± = \ ("out ± uin) (32.10) 118 Those modes that behave like positive frequency Minkowski modes in the out region (77 —• 00) are given by the asymptotic form u°kut{r},x) -> -—- exp(ikx - ioj0utV) (32.11) v/(47rw o u t) As we can see ukn and ukui are not equal, however it is possible to express u\n in terms of u°kui using a Bogolubov transformation and the linear transformation properties of hypergeometric functions, namely uln(V,x) = ak u°kut{n,x) + Bk u^'{r,,x) (32.12) where a k U „ ) 2 r ( - ^ ) ) r ( i - ( z ^ ) ) u . , m - ( i ^ ) ) r ( i ^ ) ft = (?r)* i v . - ^ n + r W ( 3 2 - 1 3 ) <^ in r ( i — ) i \ i  (i—)) We can express the amplitudes in terms of hyperbolic sinh functions, namely s i n h 2 ( ^ ± ) ak = — s i n h ( ^ - ) s i n h ( ^ w ) 2 / TIUl- • | 2 _ v P s i m V ( ^ ) s i n h ( ^ - ) s i n h ( ^ - ) (32.14) In the way of interpretation, in the remote past, where the spacetime is flat, all inertial particle detectors will register no particles, so that unaccelerated ob-servers there would identify the quantum state with the real vacuum. In contrast, in the remote future region (n -> +00) unaccelerated particle detectors there will register the presence of quanta where the expected number is given by | Bk |2-We can interpret this as the quantum creation of particles into the mode k which has come about as a result of the isotropic expansion of the universe. This is a very enlightening, albeit naive model of particle creation due to the expansion of the universe. 119 Chapter 33 Particle Creation in a Four-Dimensional Model Universe We have seen in the previous section that for a scalar field satisfying the equation d2 ^ 2 Xk+u2k(ri)xk=0 (33.1) where ul = (k2 + C{n)m2) (33.2) we have particle creation due to the evolution in the conformal scale factor C(V). The scalar field equation is similar in form to that of a quantum harmonic oscillator with a time-dependent frequency. Such equations can be solved by the W K B method in the adiabatic limit, or by any of the methods outlined in this thesis. Consider an asymptotically non-static four-dimensional model with the con-formal scale factor C(r/) given by [18] C(rj) =a2 + b2 rf (33.3) with - c o < n < oo, and where a and b are constants. 120 In the asymptotic reqions n —• ±00, the model is equivalent to the radiation-dominated Friedmann model a(t) -> y/C(t) oc s/t. In the adiabatic approximation we see that the zeroth order approximation is valid when ul{n) = [k2 + m2 {a2 + b2 rf )} (33.4) for large values of 77 or large mb, or for large values of A where A is given by A = fc2 + f ° 2 (33.5) mb The zeroth order solution in the large A limit is , 0 eM-ir)V(mb\)] (33.6) (2moA)" where 77 is fixed. While the exact solution can be expressed in terms of the parabolic cylinder function D let us look at the asymptotic distant past and distant future approx-imations: In the large | n | limit the zeroth order solution approaches XT - — f - r r e x p ^ ] (33.7) (2mo I 77 I) 2 * which is equal to the exact solution nk"^ as n -> - 0 0 and f]k°ut^ as 77 - 4 00. Solving for the Bogoliubov coefficient yields / ? , | 2 = e x p [ - 7 r ( ^ + ^ - ) ] (33.8) 121 Spectrum of a Non-Relativistic Gas A closer look at | fik | 2 tells us that this spectrum is the same as that for a non-relativistic gas of particles with momentum (33.9) at a chemical potential of and temperature (33.10) (33.11) 2nCkb where fcf, is the Boltzman constant. In the next section we shall look more closely at this spectrum temperature. 122 Chapter 34 Uniform Acceleration in Rindler Space In the case of a scalar field the Lorentz Invariant spectral function is 7T2/0(u;) = \hc2^ (34.1) Consider a scalar field of the form $(r,i) = J d3kf0(uj)cos(ik*r-iut + iQk) (34.2) The time average value of the amplitude of the field is given by the correlation function < *(0,t) * *(0,t) >= 1/2 Jd3kf0{oj) (34.3) which is Lorentz Invariant. Now consider an accelerating frame of reference moving along the x^axis with uniform acceleration a (Rindler Space) where X{T) = c2/a cosh(ar/c) (34.4) and V(T) - c tanh(ar/c) (34.5) where 7 = - v2/c2) = cosh(ar/c). 123 It is also straightforward to show that u>i = w cosh(ar/c) — ckx sinh(ar/c) (34.6) and that kxi = kx cosh(aT/c) — w/c sinh(ar/c) (34.7) The transformed correlation function is < $(0, *)* $(0, «) >accelerated= -{h/TtC){a/2c)2 COSh 2(OT/C) (34.8) Compare this to the scalar correlation function of the system at rest in a Zero Point Thermal Field: < $(0,ty $(0,t) >zPThermai= - ( fy 'KC){TTKT ' / 'h) 2 cosh 2 (TTKTT/H) (34.9) If we compare the two correlation functions we find that they are identical in functional form provided the acceleration and temperature are related by the expression T = ha/(2TtKc) (34.10) This relation is known as the Unruh-Davies Temperature relation. [27] Now consider the Lorentz Invariant spectral function for an electromagnetic field, 7r 2# 0 2(w) = l/2hu (34.11) The transformed electromagnetic correlation function is of the form ( i j = 1,2,3): < Ei(0,t)Ej(0,t) >=< Bi(0,t)Bj{0,t) > = 6ij4h/(TTc3)(a/2c)4 4 (a r / c ) (34.12) where the cross terms are of the form < Ei(0,t)Bj(0,t) >= 0 (34.13) 124 (csch is the hyperbolic cosecant). Compare this to correlation function of the system at rest in a Zero Point Thermal field, < Ei(0,t)Ej{0,t) >=< Bi(0,t)Bj-(0,t) > = 6i3Ahl{irc3)(irKT/h)4 (esc4 (nKT/h) + 2/3 C S C 2 ( T T KT/h)) (34.14) where again the cross terms are of the form < Ei(0,t)Bj(0,t) >= 0 (34.15) Notice the additional term. The question is how to interpret the functional form and the additional term. In the case of an electromagnetic field, the spectrum seen by the detector accelerating through a Zero Point electromagnetic field is n2Hlccel(co, a) = 1 /2M1 + (o/cu;)2) coth(7rca;/a) (34.16) If we express the acceleration in terms of the relation T = /ia/(27rfc(,c) (the Unruh-Davies Temperature) then the spectrum as seen by the accelerated sys-tem is ir2Hlccel{u, a) = 1 /2M1 + {IvKT/hu)2) coth{hu/2KT) (34.17) rather than the unaccelerated Zero Point Thermal spectrum n2H2atr,est(uj,0) = l / 2 ^ c o t h ( ^ / 2 ^ T ) (34.18) Note that acceleration adds a new term to the Zero Point Thermal spectrum and that the two spectrums agree at the higher frequencies hu 3> KT. In an accelerating frame there is an event horizon in the sense that in cer-tain directions events occuring beyond a certain distance from the observer can never be reported to the observer by light signals due to dilation. 125 The observer is running away with ever increasing speed from these space-time events and modulated light signals carrying information can never catch up with the observer. These modes are frozen out and the spectral distribution of eigenvalues change. A careful study of the situation shows that it is the long wavelength electro-magnetic waves that are cut-off by the event horizon. As a result the accelerated spectrum Haccei does not go over to the energy equipartition at low frequency found with the unaccelerated Zero Point Thermal spectrum HatTest-Given that acceleration and temperature are related by the expression (2nkb c) at the event horizon of a Schwarzchild black hole we have an acceleration of [21] a=4GM <34-2°) which means that an observer will find that a black hole will produce particles at a temperature of T ~ (SvGMh c) ( 3 4 ' 2 1 ) This effect is known as Hawking Radiation. In terms of the gravitational field, the vacuum around a black hole becomes unstable at the Schwarzchild radius and particles are produced with a thermal spectrum. 126 Chapter 35 Is Gravity an Induced Quantum Effect In a 1968 paper Andrei Sakharov roposed [26] that gravity is not a separately existing fundamental force but rather an induced effect associated with fluctu-ations of the vacuum state. Sakharov's Proposal is discussed in greater detail in an appendix to this thesis and in a paper by the author. [20] Einstein's Principle of Equivalence requires a modification to the Zero Point Field due to gravitational mass. Following the reasoning set out by Sakharov and Puthoff, in this section we shall show that gravity is not a separately exist-ing fundamental force but rather an induced effect associated with Zero Point Fluctuations of the vacuum. Consider the equation of motion for an oscillating charged particle given by where q — q(t) is the oscillator coordinate, vo is the natural frequancy of the oscillator, T is the damping coefficient Now consider the kinetic energy Wkin of the particle motion due to fluctua-tions induced by the Zero Point electromganetic field, u / 1 d 2 q - 1 ,dPs2 „ r o> W k i n -2m~0dp- (12*reoc3) W ( d 5 ^ 127 where p = eq is the dipole moment of the oscillator. Written in this form it is worth noting that the energy equation refers to the global properties of the oscillator (p, VQ and the damping constant T) and does not involve individual properties such as mass or charge. Using the Zero Point Electromagnetic fields E z p and B z p and solving for the time average value for < (d/dtp — x)2 > yields < (d/dtpx)2 > ~ 6e 0 c 3 / i ( rw c ) 2 (35.4) where UJC is some characteristic frequency. In two-dimensions the particle motion due to fluctuations induced by the Zero Point electromagnetic field is < (d/dtp)2 >two-dimen= 2 < (d/dtp)2 >0ne-dimen (35.5) The time average value for the internal energy of the oscillator, expressed in terms of its global properties is given by < Energy >= hTu)2C/-K (35.6) The energy calculated in this fashion is a transverse self-energy of the particle motion due to fluctuations induced by the Zero Point electromagnetic field. Using the expression Einstein expression E = mac2 gives m G = HTw2/(*<?) (35.7) In Puthoff's interpretation of Sakharov's Proposal, the oscillator's mass is of dynamical origin, originating in the motion response of the charged particle to the motion induced by the Zero Point electromagnetic field. It is the internal motion of the charged oscillator that contributes to the effective mass of the os-cillator through the mass-energy equivalence outlined in the Einstein expression E = mc2. The lowest order interaction between a charged particle and a Zero Point Field that produces a far field effect is the dipole interaction. Of the dipole-field terms, the 1/r 4 term predominates at large distances. In expanding out the dipole field distribution there is a term proportional to 1 / r 2 which is the radiation field associated with the Zero Point Fluctuation driven dipole. This radiation just replaces that being absorbed from the back-ground on a detailed-balanced basis. 128 The energy density Awd in the two-dimensional far field dipole-field inter-action is Awd = (37icr 2 cos 2 0)/(27r 2 r 4 ) / du u (35.8) Jo where wc is a characteristic frequency used as a cut-off frequency to avoid divergence. Averaged over the net contribution of randomly oriented individual Zero Point particle motion, and integrated over the solid angle, we have an overall spectral density of A ^ / = w ( ^ r 2 ) / ( 2 7 r 2 r 4 ) (35.9) Using the relationship for mass mg and F we have Apd' = u{cf )/(27r 2o; 4r 4) (35.10) Recall the expression for the accelerated Zero Point Thermal spectrum for the electromagnetic field and set T — 0 ir2H2accel{w,a) = 1/2M1 + (a/c^) 2 ) (35.11) Mult iply this expression by the density of normal modes (LJ2/TT2C3) and equate the contribution from the acceleration term l/2fkj(a/cu)2 to the ex-pression Apd' yielding ha2/{Tr2c5) = (c5m2G)/(fujy) (35.12) Now let a = Gma/r2 and solve for uc UJC = y/(ixcb/hG) (35.13) On the basis of heuristic and dimensional considerations Sakharov proposed that a vacuum fluctuation model for gravitation would have a characteristic cut-off frequency UJC of this form. Solving for the gravitational constant G we have G = Trch/huj2 (35.14) 129 Studying the Zero Point fluctuation induced dipole field at the position of particle A due to the fluctuating motion of a second similar particle, particle B , leads to an expression for the potential energy of the interaction of the form: U = -9hc3r3/(4ir) Re( f ' du (exp-2uR)/R) (35.15) Jo where u = —i u/c and uc = —i UJC/C. For two-dimensional Zero Point dipole motion the attracting potential is given by U = - 1 / 2 6(1 - cos(2R))/R3 = -8/R((sinR)/R)2 (35.16) where the parameter 6 is given by S = hT2u3/ir (35.17) and the scale parameter R is R = rujc/c. With the gravitational potential thus defined, the gravitational force is given by the classical expression Fg = -dU/dr (35.18) The gravitational potential has the desired 1/r dependence modified by a form factor ((sin R)/R)2 which has a characteristic length on the order of the fundamental length AEG — 2.82 x 10 - 3 4 cm. If we extract the leading terms from both U and Fg we arrive with the following: U = -hcr2u2/(nr) (35.19) FG = -hcTlu2cl(-Kr2) (35.20) 130 Using Sakharov's characteristic cut-off frequency ,7TC5 W c = V i hG] ( 3 5 ' 2 1 ) and the expression for the gravitational mass derived above mo = ^ £ (35.22) we arrive at the familiar expression for the gravitational force, FG = (35.23) which is Newton's Law of Gravitational Attraction between two bodies of similar mass. For dissimmilar masses we modify the force equation FG to read F g = (35.24) and solve in a similar fashion to arrive at FG = - ° ( m 2 i m 2 ) (35.25) which is Newton's Law of Gravitational Attraction between two bodies of dissimilar masses. A more detailed discussion of the consequence of this proposal is outlined in the paper Zero Point Fluctuations and the Suspended Charge Paradox [20] which is reproduced in abridged form as an appendix to this thesis. 131 Part VI Summary and Conclusions 132 Chapter 36 Summary and Conclusions Exact solutions for the Schodinger equation are known for a rather restricted set of interaction Hamiltonian or potentials, so the standard problem we are faced with is to find a good approximation in place of an exact solution. Many of our descriptions of quantum systems have been influenced by the Quantum Harmonic Oscillator (QHO). For quite a wide range of quantum sys-tems, it is valid to look for an initial approximation in the form of an oscillator basis, that is, a stable quantum system in a well chosen representation can be described by some set of harmonic oscillators with a spectrum of frequencies. Many systems may be treated as a set of oscillators with a frequency defined by a mass parameter. The interaction does not change the oscillator nature of the underlying quantum field, but only redefine their masses and other physical characteristics. The first four sections of this thesis build up the mathematical tools, namely the Algebraic Method, the Bogoliubov transformation and the "dressed" oscil-lator approach, for Part Five in which I look at Particle Creation in two Simple Models of a Friedmann-Robertson-Walker Universe as well as a hypothesis that Gravity is an Induced Quantum Effect. 133 Part VII Bibliography 134 Bibliography [1] Infeld, L . and Hull , T . The Factorization Method. Rev. Mod . Phys, 23,p. 21 - 68, 1951 [2] Dirac, P . A . M . , The Principles of Quantum Mechanics, Fourth Edition. Clarendon Press, Oxford, 1991 [3] Pena, L . and Montemayor, R. Raising and Lowering Operators and Spectral Structure: A Concise Algebraic Technique. A m . J . Phys, 48,p. 855 - 860, 1980 [4] Newmarch, J .D. and Golding, R . M . Ladder Operators for Some Spherically Symmetric Potentials in Quantum Mechanics. A m . J . Phys, 46, p. 658 -660, 1978 [5] Landau, L . D . and Lifshitz, E . M . Quantum Mechanics, Volume 3, Third Edition. Pergamon Press, Oxford, 1977 ( §112. Motion in a Uniform Mag-netic Field) [6] Rowe, P. Classical Limit of Quantum Mechanics, (Electron in a Magnetic Field. A m . J . Phys, 59,p. 1111 - 1117, 1991 [7] McMil lan , M . Quantum Leaps and Bounds, Physics 500 Notes. University of British Columbia, 1998 [8] Kaempffer, F . A . Concepts in Quantum Mechanics. Academic Press, New York, 1965 [9] Gasiorowicz, S. Quantum Physics, First Edition. John Wiley Sons, New York, 1974 [10] Morse, P . M . and Feshbach, H . Methods of Theoretical Physics, Volume 1. McGraw Hi l l , New York, 1953 [11] Schweber, S. QED and the Men Who Made It. Princeton University Press, Princeton, N . J . , 1994 [12] Bjorken. J .D. and Drell S.D. Relativistic Quantum Mechanics McGraw Hill Co, 1964 135 [13] Avery, J . Creation Annihilation Operators, McGraw Hi l l , New York, 1976, Appendix A . Kit te l , C. Quantum Theory of Solids. John Wiley Sons, New York, 1963 Louisell, W. Radiation and Noise in Quantum Electronics. McGraw Hi l l , N Y , 1964 Andion, N.P. , Malbouisson, A . P. and Neto, A . M . A n Exact Approach to the Oscillator Radiation Process in A h Arbitrary Large Cavity, Physics 0001009, 5 Jan., 2000. Gol'dman, I.I. and Krivchenkov, V . D . Problems in Quantum Mechanics. Dover Publ. , New York, 1993 Birrell . N . D . and Davies P . C . W . Quantum Fields in Curved Space. Cam-bridge Univ. Press, 1989 Boyer, T . Thermal Effects of Acceleration through Random Classical Radi-ation. Phys. Rev. D 21, p. 2139-2143, 1980 Bruskiewich, P. Zero Point Fluctuations and the Suspended Charge Paradox Gamma Magazine, Vol. 120, p. 25 - 47, Niels Bohr Inst., Copen. Denmark, 2000 Greiner, W . Muller, B . and Rafelski. J . . Quantum Electrodynamics of Strong Fields. Springer-Verlag, Berlin, 1985 Marshall, T. W . Proc. Camb. Phi l . Soc , 61, p. 540, 1963 also T. Boyer, Phys. Rev., 182, No.3, p. 1376, 1969 Puthoff, H . E . Ground State of Hydrogen as a Zero Point Fluctuation Determined State. Phys. Rev. D 35. p. 3266 -3269, 1987 Ibison, M . and Haisch, B . Quantum and Classical Statistics of the Electro-magnetic Zero Point Field. Phys. Rev., A 54, p.2737-2744, 1966 Puthoff, H . E . Gravity as a Zero Point Fluctuation Force. Phys. Rev. A 39, p.2333-2342, 1989 Sakharov, A . Vacuum Quantum Fluctuation in Curved Space and the The-ory of Gravitation. Sov. Phys. Dok., 12, p. 1040-1041, 1968 Unruh, W . Notes on Black-Hole Evaporation. Phys. Rev. D 14, p. 870-892, 1975 Welton, T. Some Observable Effects of the Quantum Mechanical Fluctua-tions of the Electromagnetic Field. Phys. Rev., 74, p 1157, 1947 A . Rueda, Behaviour of Classical Particles Immersed in the Classical Electromagnetic Zero-Point Field Phys. Rev. A , 23, p 2020 - 2040, (1981) 136 [30] Zhitnitsky, A . 500 and 501 Lecture Notes University of British Columbia, 1999/2000 137 Part VIII Appendices 138 Appendix A Some Observable Effects of Zero Point Fluctuation The quantum mechanical Zero-Point variation of the field in empty space gives rise to fluctuating electric and magnetic fields [28] whose mean square value at a point in space is given by /•OO < E2 >AV=< B2 >Av= 2hc/7r / dk k2 ( A . l ) Jo where k refers to the wave number of the quanta of electromagnetic energy. The contribution from the frequency to the mean square fluctuation in the range c dk is explicitly outlined. The mean square value of the electric and magnetic field at a point in space is derived by ascribing to each normal mode of the radiation field an energy which is just the Zero-Point energy of an ocsillator with the frequency of the normal mode. The total energy can be written either as a sum or as an inte-gration of the electromagnetic energy density. To describe the non-relativistic motion of a free charged particle, such as an electron, in a fluctuating Zero-Point field, let q be its position vector, then md2/dt2q = eE (A.2) where e is the charge of the electron and E is the fluctuating electric field whose mean square value is given above. We have disregarded any damping or back reaction in the motion of the fluctuating electron. 139 Since this equation of motion is linear we can regard it as a classical equa-tion for the quantum mechanical expectation value of the position of the electron given by q. For a given harmonic component of E (i.e. E(uj)ewi) the solution of the equation of motion is straightforward. Fluctuation in the Position of the Particle Performing this integration, find the value for < q2 >Av for the given har-monic component and sum over the frequencies using the mean square value given above. You arrive at an expression for the mean square fluctuation in the position of a free electron given by < (Ag) 2 >Av= (2/TT) a(h/mc)2 f dk/k (A.3) where a = (e2/hc). In this integral for the mean square fluctuation in the position of a free electron we have assumed both a lower cut-off fco and an upper cut-off K = mc/h. The Lamb Shift The magnitude for this mean square fluctuation in the position of a free electron will be very small for most values to the lower cut-off fco, however an observable effect known as the Lamb shift arises when an electron moves in a potential with a large curvature such as the ground state of a hydrogen or he-lium atom. Consider the motion of an electron in a statis field specified by a potential V(q). The position of the electron consists of two contributions, one from the orbital motion plus a second smaller contribution which fluctuates randomly in time. Let q be the orbital part and Aq be the random part, then the instantaneous potential energy V(q + Aq) is given by V(q + Aq) = [1 + A g • V + l / 2 ( A g • V ) 2 + ---] V(q) (A.4) The effective potential energy that the electron sees is the average of V(q + Ag) over all values of A g . Since A g has an isotropic spatial distribution we get <V(q + Aq) >Av= [1 + 1/6 < (Ag) 2 > A v V 2 + • • • ] V(q) (A.5) 140 As can be seen, the existence of the fluctuations in the position of the electron effectively modifies the potential in which it moves by adding a term propor-tional to the Laplacian of the potential energy. Consider the perturbation to the ground state potential energy of a hydrogen atom caused by the fluctuations in the position of an orbiting electron. Given the static potential of the nucleus V(q) = -(1/47TCO) (e 2/V) (A.6) Retaining the first order term in the effective potential energy, the correction to the potential energy becomes AV(q) = (e2/37re0) a (h/mcf In [mc2/hcko]6(q) (A.7) The correction to the energy W of the stationary state is given by AW = (e2/37re0) a (h/mc)2 In [mc2/hck0] | *(0) | 2 (A.8) By letting the quantity hcko equal to the average excitation energy of 17.8 Rydbergs for hydrogen this expression becomes identical with the expression derived by Bethe for the level shift in hydrogen. Reviewing the equation it becomes obvious that the fluctuation in the posi-tion of the electron must act to weaken the effect of the potential energy. This results in an observable shift in the energy levels which were first measured some fifty years ago at Cornell University. Low Energy Compton Scattering The fluctuations in the position of the electron results in a spreading out of the electronic charge and the current that participates in electromagnetic interactions. Consider the modification to the transition probability in non-relativistic Compton scattering due to the fluctuations in the position of the electron. Under the non-relativistic Compton scattering a free electron executes a steady forced oscillation under the action of an incident light wave and emits a scattered light wave of the same frequency. The effect of the position fluctuation is twofold. The electron behaves like a distributed charge with a mean square radius < (Ag) 2 >Av It therefore interacts less strongly with the incident light and radiates a weaker scattered wave than what would be expected from a point particle. 141 To estimate the magnitude in this reduction in the interaction between the electron and the incident light we consider the change in the phase factor for the wave introduced by the averaging over the position fluctuation of the electron. This yields < exp[ik • (q + Aq)] >Av= exp(ik • q) < [1 + Aq • V + l/2(Aq • V ) 2 + • • • ] >Av (A.9) which can be reformulated to read < expik • (q + Aq) >Av~ exp{ik • q) [1 - 1/6 k2 < (Aq)2 >Av] (A.10) The correction involves the product of the mean square position fluctuation with the Laplacian of the space function describing the interaction. We see that the amplitude of the oscillation and the amplitude of the scattered wave will suffer a fractional reduction equal to The resultant reduction will be twice as large and the reduction in the scat-tered intensity or cross section will be twice as large again , so that the fractional change in the Compton scattering cross section will be Aa/a = -2/Zk2 < (Aq)2 >Av= 4/(37re0) a(hk/mc)2 In [mc/hk0] (A.12) We observe that the angular distribution will remain unchanged since the scattering retains its dipole character. There remains the problem of determin-ing the lower cut-off ko-Frequencies of fluctuation higher than the frequency of the incident light wave will effectively spread out the scattering charge, while frequencies below this limit will only displace the scattering charge in a random fashion, therefore set ko = k to get - 1 / 6 k2 < (Aq)2 >A ( A . l l ) Aa/a = 4/(3Treo)a(Hk/mc) 2 In [mc/hk] (A.13) Note that the correction goes to zero strongly at low frequencies. 142 The Interaction Between a Spin and a Magnetic Field Fluctuations of the radiation field change the effective potential energy of interactions between an external magnetic field and the spin and orbital mag-netic moment of an electron. Consider a simple system which consists of an angular momentum with an associated magnetic moment. Let 1/2/ur be the angular momentum operator. Then the equation of motion for the angular momentum operator is given by d/dt a = (e/mc)B x a (A.14) where B is the instantaneous magnetic field intensity arising from the fluctu-ations in the Zero-Point field. The correction ACT to the angular momentum operator is given by ACT = (e/mc)B x a (A.15) The mean square fluctuation in the unit spin operator < (ACT) >AV is then given by < ( A C T ) 2 >Av= a2a2/Tr(h/mc)2 f dkk (A.16) Jo The upper limit to the integral has been made finite rather than infinite. We define the mean square angle of fluctuation < ( A 0 ) 2 >AV by the expression < ( A O ) 2 >Av=< ( A C T ) 2 >AV /O2 = a/7r(h/mc)2K2 (A.17) The energy W of a particle with spin in a magnetic field is given by W = {eh/2mc) | CT | cosQ where 0 is the angle between the spin direction and the direction of the external magnetic field. It is immediately apparent that the fluctuations in the magnetic field affects the average value of cos 0 . This means that cosQ is to be replaced by an average value < cos 0 >AV which is taken as an average over the fluctuations in the spin vector. It can be shown that < cos 0 >Av=< cos 0 cos A 0 -I- sin 0 cos 4> >AV (A.18) 143 This average is easy to assess given that the average of cos<j> must vanish because of the isotropy of the fluctuation. Given that A 0 is small, cos A 0 can be respresented by the first two terms in its series expansion so that < cos 0 >Av = < cos 0[1 - 1/2 < ( A 0 ) 2 >AV] >=< cos 0[1 - a/27c(hK/mc)2 > We see that the correction to the orientation energy of the orbital spin consists in a reduction in the magnitude of the energy proportional to the energy itself. It is convenient to consider the effect of the interaction of the orbital spin with the radiation field as consisting of an alteration of the magnetic moment of the electron \i. If we set K = l/a = aimc/h) then which is a very small effect. For the intrinsic spin of the electron the alter-ation of the magnetic moment of the electron /x is given by (A.19) A/x/M - (1 /2TT) a 3 (A.20) An/fi = - ( 1 / 2 T T ) a (A.21) 144 Appendix B Zero Point Fluctuations and the Suspended Charge Paradox From electrodynamics we know that an accelerating charge with colinear veloc-ity and acceleration radiates with the following angular dependence: [20] dP _ e2a2 s in 2 6 dQ ~ (47rc3(l - 8 cos0) 5) ( ' ' where a is the acceleration, 8 = v /c and 8 the angle measured in the direc-tion of motion. In the limit of small velocity and integrated over the solid angle the total power radiated by a nonrelativistic accelerated charge is given by the familiar Larmor formula: Einstein's Principle of Equivalence tells us that an accelerating and a grav-itational frames of reference should be equivalent on a local scale. The "Paradox of the Suspended Charge" arises when we set a = g, the ac-celeration due to gravity, then we would expect the suspended charge to radiate away its power at a rate proportional to g2, namely P ~ e2g2/c3 (B.3) something we of course do not see. 145 If as it appears that an electron has no finite size then the "Paradox of the Suspended Charge" carries down to a very small length scale. What is Zero Point Fluctuation? The random radiation involved in Zero Point Fluctuations is not connected with temperature radiation, but exists in the vacuum at the absolute zero of Thermal Temperature. These fluctuations are a result of the Heisenberg Un-certainty relation AEAt > h, although the underlying physical cause is yet not fully understood. This random radiation is considered as real as thermal radiation. Some of the observable effects of this Fluctuation is outlined in an appendix to this thesis. A special aspect of Zero Point Fluctuation is that its spectrum is Lorentz Invariant. This means that for a given field type every inertial observer, irre-spective of their velocity, finds the same spectrum for the Zero Point Field. [22] For a scalar field the Lorentz Invariant spectral function is given by: [19] For an electromagnetic field the Lorentz Invariant spectral function is given by the familar equation: Stability of Ground State One of the great questions of quantum and classical physics is why atoms do not radiate away all their energy and collapse down to zero. It appears that Zero Point Fluctuation prevents the collapse. To describe this effect requires both a classical and a quantum description of the process of absorption and radiation from an atom. [23, 24] Define the relationship between the quantum expectation value and the clas-sical probability by the following: r r 2 / o M = \hc2 (B.4) 7T2/l0(w) - -HUJ (B.5) (B.6) 146 Let f(r) — exp(—is * r) so then < $ | e x p ( - i s * r ) | * >= jdTr P*(r) exp( - i s * r) = g(s) (B.7) where g(s) is a generating function. So then l/(27r) r a j dTr g(s) exp(-is *r)= P*(r) (B.8) where m is the dimensionality of the space. You recognize this relationship to be a fourier transformation from the r-space to the s-space. The nonrelativistic equation of motion for an oscillating charged particle is given by g+ * - r £ f - r * (B.9) where q = q(t) is the oscillator coordinate, vo is the natural frequancy of the oscillator, T is the damping coefficient e 2 (67re 0m ec 3) (B.10) and T* is the driving coefficient T* = — ( B . l l ) Fourier transform the equation of motion and solve for q(u) q(v) = H{v)E{v) (B.12) where the dispersion relation H(v) is given by H ^ = 7~2 TT^Fln <B-13) (VQ — + iTv6) Describe the zero point electric field E z p by a traveling wave Ezp(r,t) = ReYekO-kWk exp(ik * r - iut) (B.14) 147 with wk = uk + ivk, tk a unit vector in the direction of propagation and ak is the polarization. We shall introduce the intensity of the E z p field by letting uk = (y/h) cos(0fc) and Vk = (\Afc) sin(Qfc), where 0* is a random phase subject to a constant dis-tribution in [0,27r], and Ik is the intensity per mode. Set r=0 (to suppress the k*r term in Ezp(r,i) ) and taking the Fourier transform of EZP(r, i) yields Ezp(u) =nYl e^k(S{uj + v){uk + ivk) + (6(CJ - u)(uk - ivk)) (B.15) So then solving for q(t) q{t) = l /(27r) m j dv exp{iut) H{u)E{v) = ^ e ^ f l e C ^ C M ) (B.16) where £ ( C J ) = exp(iuit)H(w). The generating function g(s) =< $ | exp{-is * r) | $ > for this distribution is given g(s) = n*(l /(27r) m ) jduk jdvk exp((-w f c /2 - vk/2) -(is * tkak(uk fle(C(w)) + wfcflc(C(w)))) (B.17) or g{s) = nk exp( - | s * ekak H(u) | 2 /2) (B.18) It is worth noting that this result is exact and that there has been no re-quirement for h (Planck's constant) up to this point. In an unbounded space (L ^> A), the mode product 11* in the generating function g(s) can be converted to an integral within the exponential, so that ^ ) = e x p [ - I ^ | ? ] JduLj3\H(U)\2) (B.19) where the dispersion equation is given by 148 Notice that Planck's constant h has been introduced for the first time in the generating function and is being used as a scaling factor. Reformulating the dispersion equation ( n 2 ((w 2-u;2)2 + (ru,3)2) l2 = u.a . « , m , W ( B - 2 1 ) Due to the smallness of the damping term F in the denominator the inte-grand is sharply peaked at w = wo-Using the resonance approximation /•oo roc -_'p*2 J du w 3 | H(u) | 2 ~ y_ dw wl | H(wo) | 2 * ^ r j j y (B.22) Then . 2 ^ . „ 3 /•OO Q^-2, „3 / d w u 3 | H(u>) | 2 ^ (B.23) Jo (rnewo) The generating function in the resonance approximation is <,2fc „ 2 „ 2 \ ^ e x p ( - - ) = exp(- S 2 <r 2 ) (B.24) 4meu)o and the probability distribution becomes ™ = ( ^ - P ( - & ) (B.25) where a2„ = „ fi,, . 9 2mecjo Note that this distribution agrees in form with that predicted by quantum mechanics for the nonrelativistic harmonic oscillator in the ground state. If we consider the ground state of the Bohr atom as modeled by a pair of orthogonal one-dimensional harmonic oscillators then the two-dimensional distribution becomes: 149 Compaxe this with the quantum probability for a two-dimensional quantum oscillator in the ground state meu0 moj0{x1+y2) Po(x,y) = (-^7-) exp[ ] (B.27) and note that they have similar functional form. We have derived a quantum mechanical ground state probability distribu-tion for a simple atom using classical field theory. The underlying mathematics of the Gaussian function and the fact that a Gaussian function carries over to a Gaussian function under a Fourier transform is worth remembering. Power Absorbed from the Zero Point Field Now look at the time-average power that is being absorbed from a Zero Point electric field E z p bv a harmonic oscillator: < Pabs >=e< E z p *v> (B.28) where E z p and B z p (the Zero Point magnetic field) are given by EZP(r,t) = Re"Yd3keky/Ik exp(ik *r-iu>t + i@k) Bzp(r,t) = Re^^H* x ek)Vh exp{ik * r - iut + i@k) (B.29) Let us now use a classical intensity function Ik - 2£ - s f e ™ where you recall /IQ(W) is the Lorentz Invariant spectral function for the vec-tor Electromagnetic field. The time average power absorbed by the oscillator is then r/l f°° Tu <Pabs>=(-)Jo ^ ( ( w 2 _ u ; 2 ) 2 + ( r a ; 3 ) 2 ) (B.31) This integral is strongly peaked at LJO so then the resonance approximation can be used and the integral becomes < ^ > g l / 2 £ ^ .^/f (B.32, 150 This integral describes a Lorentzian line shape and is equal to TTU>Q, so then < P a 6 s > ^ — - 1 2 7 r e o m e c 3 (B-33) This is an expression for the absorption of electromagnetic power from the Zero Point field by a one dimensional charged harmonic oscillator. Model the ground state motion of the Bohr atom with radius TQ by a pair of one dimensional harmonic oscillators describing circular motion. The time averaged electromagnetic power absorbed for a two dimensional oscillator is then simply ^ Plbs -^two—dimen— 2* < P^one—dimen (B.34) The time averaged electromagnetic power radiated by an electron in circular motion with acceleration a is given by the well known expression < Prod >= (B.35) Now consider the ratio of the time-average radiated power to the time-average absorbed power, < Prad > I < Pabs >= Tnerlcj0/fi (B.36) We know that the ground state of an atom constitutes a stable state. We now also see that it is a state that is in a dynamic balance between the elec-tromagnetic energy that is being radiated into the Zero Point Field and the electromagnetic energy that is being absorbed from the Zero Point Field, such that < Prad > = merlu0 = 1 < Pabs > n We recognize this as the angular momentum quantization condition first introduced by Niels Bohr in 1913 merlu)0 = n h (B.38) with n = 1. As outlined, atoms do not collapse down to zero size due to a detailed-balance between the electromagnetic energy that is being radiated into the Zero Point Field and the electromagnetic energy that is being absorbed back from to the Zero Point Field. 151 Zero Point/Thermal Spectral Function Up until this point thermal temperature has not been included in the spectral function. Let us now add Planck's thermal temperature spectrum to the Zero Point electromagnetic spectrum: TT2H^(UJ,T) = 1/2 fej + hu(exp{huj/kBT) - 1) (B.39) where T is the thermal temperature and K is Boltzman's constant. We can reformulate this spectral function to read 7T 2# 0 2(w,T) = 1/2 hucoth{l/2fkj/BT) (B.40) which is a combined Zero Point/Thermal Spectral Function. Unruh-Davies Temperature In the case of a scalar field the Lorentz Invariant spectral function is as outlined above, namely [27] * 2 / o M = • (B.41) Consider a scalar field of the form $(r, t) = Jd3kf0(uj) cos(ik*r -iwt + iQk) (B.42) The time average value of the amplitude of the field is given by the correlation function (a correlation function is to classical field theory what an expectation function is to quantum field theory) < $(<),<) * #(0,i) >=\j d3kf0(uj) (B.43) which we know to be Lorentz Invariant. Now consider an accelerating frame of reference moving along the x-axis with uniform acceleration a. It can be shown that X(T) = — cosh(ar/c) (B.44) a and that 152 V(T) = c tanh(ar/c) (B.45) where 7 = ^ / ( l - v2/c2) = cosh(aT/c). It is also straightforward to show that cjf = w cosh(ar/c) — cfcs sinh(ar/c) (B.46) and that kxi = kx cosh(ar/c) — C J / C sinh(ar/c) (B.47) The transformed correlation function is < $ (0 ,0* *(<M) >acce/erated = - — (^") 2 COSh 2(ar/ C) (B.48) 7TC Z C Compare this to the scalar correlation function of the system at rest in a Zero Point Thermal Field: < $(0,<)* *(<M) >ZPThermal= ~ ( ^ ) 2 COSh2(7rfc67Y/ft) (B.49) 7TC n If we compare the two correlation functions we find that they are identical in functional form provided the acceleration and temperature are related by the expression T = ^ k <B-50» This relation is known as the Unruh-Davies Temperature relation. Now consider the Lorentz Invariant spectral function for an electromagnetic field, 7r 2tf 0 2(w) = l/2hw (B.51) The transformed electromagnetic correlation function is of the form (i j = 1,2,3): < Ei(0,t)Ej(0,t) >=< ^(0 ,1)5^(0 ,1) >= <5ij4ft/(7rc3)(a/2c)4 csch4{ar/c) (B.52) 153 where the cross terms are of the form < Ei(0,t)Bj(0,t) > = 0 (B.53) (csch is the hyperbolic cosecant). Compare this to the correlation function of the system at rest in a Zero Point Thermal field, < £i(0,*)£,-((),<) >=< Bi(0,t)Bj{0,t) > = SijAh/iTrc^inKT/n^icsc^nKT/h) + 2/3csc2(TrKT/h)) (B.54) where again the cross terms are of the form < Ei{0,t)Bj(0,t) >= 0 (B.55) Notice the additional term. The question is how to interpret the functional form and the additional term. A clue is to be found in the result for a scalar field and the Unruh-Davies Temperature relation. In the case of an electromagnetic field, the spectrum seen by the detector such as Casimir plates accelerating through a Zero Point electromagnetic field is 7T2Hlccel(u>, a) = 1/2M1 + (a/cw) 2) coth(7rca;/o) (B.56) If we express the acceleration in terms of the Unruh-Davies Temperature relation T — ha/(2nKc) then the Zero Point Thermal spectrum as seen by the accelerated system is 7 r 2 t f 2 c c e , (w , a) = 1/2M1 + (2TTKT/HLJ)2) coth(hu/2KT) (B.57) rather than the unaccelerated Zero Point Thermal spectrum n2Hltr.est(uj,0) = l/2fru]coth(hLj/2KT) (B.58) Note that acceleration adds a new term to the Zero Point Thermal spectrum and that the two spectrums agree at the higher frequencies hui ^> KT. In an accelerating frame there is an event horizon in the sense that in cer-tain directions events occuring beyond a certain distance from the observer can never be reported to the observer by light signals due to dilation. The observer 154 is running away with ever increasing speed from these space-time events and modulated light signals carrying information can never catch up with the ob-server. These modes are frozen out and the spectral distribution of eigenvalues change. A careful study of the situation shows that it is the long wavelength electro-magnetic waves that are cut-off by the event horizon. As a result the accelerated spectrum Haccei does not go over to the energy equipartition at low frequency found with the unaccelerated Zero Point Thermal spectrum Hatrest-Sakharov's Proposal Andrei Sakharov's Proposal is that gravity is not a separately existing fun-damental force but rather an induced effect associated with fluctuations of the vacuum state. Sakharov's Proposal was discussed in detail in a paper written by H . E . Puthoff of the Institute for Advanced Studies in Austin, Texas. [26, 25] Consider again the equation of motion for an oscillating charged particle given by d2/dt2q + v2q-rd3/dt3q = r*E (B.59) where q = q(t) is the oscilator coordinate, VQ is the natural frequancy of the oscillator, T is the damping coeeficient r = e 2 /(67re 0m ec 3) (B.60) Now consider the kinetic energy Wkin of the particle motion due to fluctua-tions induced by the Zero Point electromganetic field, W^n = l/2m0d2/dt2q = l/2(d/dtq)2 / m 0 = (d/dtp)2/(12irTe0c3) (B.61) where p = eq is the dipole moment of the oscillator. Written in this form it is worth noting that the energy equation refers to the global properties of the oscillator (p, vo and the damping constant T) and does not involve individual properties such as mass or charge. Using the Zero Point Electromagnetic fields E z p and B z p outlined above in section 4. and solving for the time average value for < (d/dtp — x)2 > yields < (d/dtpx)2 > ~ 6e 0c 3/i(rw c) 2 (B.62) where uc is some characteristic frequency. In two-dimensions the particle motion due to fluctuations induced by the Zero Point electromagnetic field is 155 < {d/dtpf >tw0-dimen= 2 < (d/dtp)2 > one—dimen (B.63) The time average value for the internal energy of the oscillator, expressed in terms of its global properties is given by < Energy >= hTuj2C/-K (B.64) The energy calculated in this fashion is a transverse self-energy of the particle motion due to fluctuations induced by the Zero Point electromagnetic field. Using the expression Einstein expression E = mac2 gives mG = hTu2cl{Trc2) (B.65) In Puthoff's interpretation of Sakharov's Proposal, the oscillator's mass is of dynamical origin, originating in the motion response of the charged particle to the motion induced by the Zero Point electromagnetic field. It is the internal motion of the charged oscillator that contributes to the effective mass of the os-cillator through the mass-energy equivalence outlined in the Einstein expression E = mc2. The lowest order interaction between a charged particle and a Zero Point Field that produces a far field effect is the dipole interaction. Of the dipole-field terms, the 1/r 4 term predominates at large distances. In expanding out the dipole field distribution there is a term proportional to 1 / r 2 which is the radiation field associated with the Zero Point Fluctuation driven dipole. This radiation just replaces that being absorbed from the background on a detailed-balanced basis. The energy density Awd in the two-dimensional far field dipole-field inter-action is Awd = {ShcT2cos2e)/(27r2r4) f ' du OJ (B.66) Jo where uc is a characteristic frequency used as a cut-off frequency to avoid divergence. Averaged over the net contribution of randomly oriented individual Zero Point particle motion, and integrated over the solid angle, we have an overall spectral density of Apdi = w(^r 2 ) /(27T 2 r 4 ) (B.67) 156 Using the relationship for mass mg and T we have APd/ = u{cf )/(27r 2a; 4r 4) (B.68) Recall the expression for the accelerated Zero Point Thermal spectrum for the electromagnetic field and set T = 0 n2H2accel(u:,a) = 1 /2M1 + (a/cuj)2) (B.69) Multiply this expression by the density of normal modes (u>2/^c3) and equate the contribution from the acceleration term l/2hu(a/cui)2 to the ex-pression Apd' yielding ha2/(7T2c5) = (c5m2G)/{hu>y) (B.70) Now let a = Gmo/r2 and solve for w c U c = y/{it(?'lhG) (B.71) .. On the basis of heuristic and dimensional considerations Sakharov proposed that a vacuum fluctuation model for gravitation would have a characteristic cut-off frequency uc of this form. Solving for the gravitational constant G we have G = nc5/hj2 (B.72) 157 The Suspended Charge Revisted Let us now look again at the contribution from the acceleration term in the expression Apdi given by Apdi = (1/2 M a 2 2 7 r V (B.73) Consider the following expression relating time average differential power to the incident flux, < DP/DO, >=< da/dQ X incident flux > (B.74) where for a massless relativistic particle < incident flux >= c < Energy > (B.75) This means that (e2(sin2<9)/(47rc3(l - /?cos0) 5 )a 2 = < da/dil >c J duiApd/ (B.76) Integrate over w to get < Energy >= / dwApdi = hu2a2/(4n2c5) (B.77) Solving for the differential cross section < da/dCl > yields < dor/dfi >= {e2/hc){hTTC2/cj2)sin2 0/(1 - /3cos6»)5 (B.78) Notice that the dependence on the square of the acceleration drops out. Replace (e2/hc) by the fine structure constant a and w 2 by the Sakharov characteristic cut-off frequency u2 and you get the following: < do/dCl >= {aGh/c3)(sm2d/(l - /3cos0) 5 ) (B.79) The angle 9 points in the direction of the gravitational gradient. This is an interesting expression in that it connects the fine structure constant a with the gravitational constant G . We shall call this expression the Electro-Gravitational differential cross sec-tion. [20] 158 The Electro-Gravitational Cross Section For 3 <3C 1 and integrated over the solid angle, the Electro-Gravitational differential cross section has the value °EG(Q < 1) - 4naGh/Sc2 ~ 7.92 x l ( T 6 8 c m 2 (B.80) This Electro-Gravitational differential cross section is many orders of mag-nitude smaller than cross sections for typical electromagnetic interactions. This exceedingly small cross section is due to gravity not being a separately existing fundamental force but rather an induced effect associated with fluctu-ations in the Zero Point electromagnetic field. A fundamental length is given by the square root of the Electro-Gravitational differential cross section, namely A EG = S/O-EG ^ 2.82 x 1 0 - 3 4 c m (B.81) which is on the order of the Planck length Kpianck — 10~ 3 3 cm. It is also worth noting that the Electro-Gravitational differential cross section indicates that if a particle does not participate in the electromagnetic interac-tion then it does not have gravitational mass. If this hypothesis is indeed correct then under the Eotvos Principle of the Equivalence of Inertial to Gravitational Mass, a particle that does not partici-pate in the electromagnetic interaction would not have inertial mass. A particle such as a neutrino is essentially massless under this proposal, what little mass it may have being a result of the Electro-weak interaction. 159 Gravitational Force Studying the Zero Point fluctuation induced dipole field at the position of particle A due to the fluctuating motion of a second similar particle, particle B , leads to an expression for the potential energy of the interaction of the form: [25] " - - T T * , A 5 - > < B ' 8 2 ) where u = —i ui/c and uc = —i u>c/c. For two-dimensional Zero Point dipole motion the attracting potential is given by ^ = _ 1 / 2 A i ^ ( ^ ) = _ A ^ ( ! | « ) 2 (B.83) where the parameter A is given by A = « M (B.84) and the scale parameter R is given by R = 2± (B.85) c where r is the distance between the two dipoles. Wi th the gravitational potential thus defined, the gravitational force is given by the classical expression Fg = (B.86) dU dr The gravitational potential has the desired 1/r dependence modified by a form factor ((sin R)/R)2 which has a characteristic length on the order of the fundamental length KEG — 2.82 x 1 0 - 3 4 c m . If we extract the leading terms from both U and Fg we arrive with the following: tf = - * * M (B.87) F o = _ M M ( B . 8 8 ) 160 Using Sakharov's characteristic cut-off frequency "e = Vi-jg) (B-89) and the expression for the gravitational mass derived above mG = ^4 (B.90) we arrive at the familiar expression for the gravitational force, which is Newton's Law of Gravitational Attraction between two bodies of similar mass. For dissimilar masses we modify the force equation FG to read , F o = z M l M (B.92) and solve in a similar fashion to arrive at F g = ( B . 9 3 ) which is Newton's Law of Gravitational Attraction between two bodies of dissimilar masses. 161 Appendix C Why the Cut-off? What could be the cause of the cut-off in u considered in Sakharov's proposal? For a particle of diameter 2R > 0 spectral components of the electromag-netic Zero-Point random radiation whose wavelength A is smaller than the size 2R of the particle cannot be effective in producing translational motion of the particle. [29] Only spectral components with wavelengths A > 2R can be responsible for translational motion of the particle as a whole. Spectral components smaller than the size of the particle can only be effective in producing internal defor-mation or rearrangements. Therefore, despite the spectral divergence of the Zero-Point spectrum, and its associated infinite energy density, a natural cut-off should appear that is related to the size of the particle. Convergence of the Form Factor A convergence form factor can be obtained by finding an upper bound to the energy available from the electromagnetic Zero-Point field for a charged particle of non-vanishing size. Model the particle as a homogenously charged sphere. Since the spectrum of the electromagnetic Zero-Point field is Lorentz Invariant we are not concerned about velocity effects and we can work in the frame of reference where the par-ticle is instantaneously at rest. Let the particle have a small non-zero volume v, with D « V , where V is the electromagnetic cavity volume and let the time duration of the interaction be a short non-zero time interval r > 0. 162 To study the translational effect of the field on the particle consider the jth field component {E3)VT = — / d3x f dtEj(x,t) (C . l ) VT Jv JT The expectation value of E2 is given by averaging over the volume of the particle and the duration of measurement so that < 0 | {Ej)lT | 0 >= r - i r j f f d3x d3x'dtdt' < 0 | Ej(x,t)Ej(x', f)\0> \VT) J V T J U T (C.2) The matrix element in the integrand is < 0 | Ej{x,t)Ej(x',t') | 0 >= y ]T^«exp( i [ fc s • {x - x') - u,(t - t'])(esX -ij) (C.3) It follows then that < 0 | {Ej)2VT | 0 > = T7 T-^ I f d3x d3x'dtdt' 2 V fkjsexp(i[ks • (x - x') - ua{t - t'UCA) V (VT)2 JV,TJV,T Carrying out the time integration and replacing the summation over s by an integration and expressing the energy density u=< E2 > /4ir and recalling that the total energy density is u = V~x l/2fajg we arrive at an expression for the average electromagnetic Zero-Point energy density over the volume of the particle, namely j&e^i.d-S!))^^ (C.5) We integrate first over k, letting the £3 axis be parallel to the vector x - x', to yield he f f f sin2(l/2cfcr) < u >= / d3x x' dkd&kd(j)kk3 sin Qk , , ' ,2 exp(ik \ x - x' | cosQ k) (27T) V Jv Jv Jk (l/ZCKT) (C.6) where the integration is taken over an infinite sphere in k-space. 163 Since 1 , i i . * „sin(fc | x — x' I) „. exp(ik \ x - x ' \ fi)dfi = 2 fc | ^ _ ^, | (c-7) we obtain 2fic /" ,, [ , , r°° sin 2(l/2cfcr) sin(fc I x - x ' I) ^ o x This equation gives a divergent expression if v = 0 or r = 0. This follows that when the particle volume is very small and the time of the interaction is very short, there is no averaging of the high-frequency components and the full field acts on the particle. In contrast, when v and r are finite, non-zero quantities, the high-frequency components give no contribution to the energy available from the field. In the case of a spherical particle it can be shown expanding out the volume integral in terms of spherical Bessel, spherical Hankel and spherical Harmonics that the energy density becomes 9fic f°° sin 2(l/2cArr) .sin{kR) „ m l 2 <** ,nn\ < U > = ( ^ P J J 0 (l/2efcr)» [ - m - C ° S ( k R ) ] T ( C - 9 ) A signal takes a maximum time 2R/c in traversing the particle. It can be assumed that the maximum time of detection approximately cor-responds to this amount, then 9hc f°° sin 2(a) .sm(alpha) , da < U > = ( 2 ^ / 0 l a j ^ - W " C O S ( Q ) ] T ( a i 0 ) where a = kR. 164 For the case R > 0 the energy available from the field, U = v < u > remains bounded, namely /•OO U = v 7(cj)p(w)dw ( C . l l ) JO 10 where p is the Zero-Point spectrum and i \ r i 9 . s i n a . 2 . s ina 7(w) = 7[a] = —( 2 cosa 2 C.12 a 4 a a where a = wR/c. A cut-off occurs around the critical wavelength A c = 2R, that is at a critical frequency " c = | (C.13) Thus wavelengths smaller than the size of the particle produce internal de-formation or rearrangements and do not directly contribute to the translational motion of the particle. Estimate of the Size of Fundamental Particles If we equate the critical frequency to the Sakharov frequency we can solve for R namelv R = Vl*-f] (C.i4) giving an estimate for the size of fundamental particles like electrons and quarks on the order of R w 2.85 x 10~ 3 3 cm (C.15) where we have assumed that constants like h~ and G are not running con-stants. It is interesting that with the Sakharov frequency we can use the Gravita-tional constant in an estimate of electrons and quark size. 165 

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