t is integrated over. The prescription for doing so is aJ2 (A-3) t=i 8 l 4>t-4>t-\ dr a a With this transformation, the partition function is z=(n\u00C2\u00A3*)\u00C2\u00BB\"iE\",[mV''J- ^ with boundary conditions 4>t+T \u00E2\u0080\u0094 t-This path integral can be solved analytically by finding the eigenfunctions of the ac-tion and boundary conditions. Here the appropriate transformation is a discrete Fourier transform given by 4H = ^=_Z**e\u00E2\u0084\u00A2klT (A.5) and inverse transform V1 t=l Applying this transformation and making use of the identity I ^ e ^ - ' W T ^ ( A . 7 ) gives that T k 1 PCX) 1 POO n / ^ =n / t'=ij-\u00C2\u00B0\u00C2\u00B0 k'=ij-\u00C2\u00B0\u00C2\u00B0 E^ 2 = Ei^ i 2 (A-9) 74 Appendix A. Analytic Calculations for Matrix Integrals and t k k = J ]4s in 2 (7r fc /T) | ^ | 2 . (A.10) Thus after the transformation, the partition function can be written as Z. f T n Kk' = lJ-\u00C2\u00B0\u00C2\u00B0 / T n \,fc'=l' ( A . l l ) Letting Ak = \ sin (nk/T) + an1, gives that n \k'=l <>k' e 2 ELI iAk*k (A.12) Since ^ is complex, this integral only makes sense if the measure is actually dfad*k. This situation is remedied by changing some of the fa modes to complex conjugate modes. Since t is real, then fa = (j)*_k is one such expression. Applying periodic boundary con-ditions gives that fa = 4>T-k- Thus the top half of the modes can be written in terms of equivalent complex conjugate modes. The path integral becomes Z = TV 2 ,oc n ^'=1 \u00E2\u0080\u00A2 \u00C2\u00BBk'ak (A.13) This expression can be solved using basic Gaussian integrals. To be thorough, it is necessary to calculate in terms of the component variables of fa. Let fa = ak + ibk, then h i t ) 1 71 1 i 1 - i \ / \ ak V b k ) (A.14) 75 Appendix A. Analytic Calculations for Matrix Integrals and (772 \ fj r dak,dbk, e-^lL2M+^. (A.15) Integration only occurs when k' = k and this eliminates the sum in the exponential. The result is the product of T/2 Gaussian integrals 3 7 2 / r\u00C2\u00B0\u00C2\u00B0 1 \ 2 = IT / dakdbke-^al+b^A\" . (A.16) k=i ' Using The result is 2 = n| (A.18> *;=i * r / 2 2. n This is the value of the partition function that can be calculated by dividing the integral into time slices. Ideally it should be possible to restore the continuum limit: a \u00E2\u0080\u0094> 0, T \u00E2\u0080\u0094> 0 0 , holding 6 fixed. A.1.2 Solving Via Functional Determinant A n alternate way to solve this problem is to leave the fields continuous and express the partition function as a functional determinant Z = yV^]e~5/o/W+MV ( A . 1 9 ) 1 d e t ^ (~W + ^2 The solution is then the product of the eigenvalues of the determinant, which are found by solving the equation ( \u00E2\u0080\u0094 + u2)ip = Xnip. The eigenfunctions that satisfy this equation are ip = e^ 2 \u00E2\u0084\u00A2//^ a n d t n i l s , Xn = (^f^ + p2 for n 6 Z. Note that these are the 76 Appendix A. Analytic Calculations for Matrix Integrals + M 2 same eigenfunctions that were needed in the previous section, except that there is now no restriction on how large n can be. As such, these two techniques are basically different ways of doing the same thing. The resultant partition function is OO -. z= n T \u00E2\u0080\u0094 : \u00E2\u0080\u0094 ^ ( A - 2 \u00C2\u00B0 ) \ P J Using a product from the Gradshteyn and Ryzhik table of integrals (p. 43) [28]: sinhz = ,n(l + ^ ) ( A - 2 1 ) n = l ^ ' and manipulating the partition function to match the form of this product, reveals that the partition function is a hyperbolic sine to within a multiplicative constant 1 (A.22) sinh( The exact result for a quantum harmonic oscillator is Z = 2 s i n h ( / 3 ^ / 2 ) , so the necessary constant in this expression is ^. This expression can also be derived using a contour integral technique but the details are not elaborated upon here. A . 1.3 2-Point Correlation Function It is often more practical to calculate the value of an observable, rather than the just the value of the partition function itself. For the one dimensional harmonic potential, an observable is calculated as (0) = I y ^ J O ^ e - U ^ + ^ V (A.23) For instance, consider the two point correlation function n 0 m ) . (A.24) 77 Appendix A. Analytic Calculations for Matrix Integrals Using the discrete Fourier transform from the previous section gives T 2iri(np\u00E2\u0080\u0094mp') JT (A.25) and (\u00C2\u00ABm>4EEe2*i(\"')/T p p' p P' \fe'=i d(f>k' (A.26) Changing the integral measure to dfadcfi^ and noting that when p ^ p' the integration is over an odd function and is thus zero, gives {4>ncf>m) = I ^ e 2 - ( \u00E2\u0080\u0094 ) P / T m i \fc'=ij I 2 P - E TII k=l ^kAk<)>k (A.27) The right hand side can be evaluated using Gaussian integrals, such as was done in the previous section and thus T/2 n . f c ' = i TT \u00E2\u0080\u0094 I \u00E2\u0080\u0094 \u00E2\u0080\u0094 }l^]ApAp i r i r (A-28) z_ Ar, After relabelling p \u00E2\u0080\u0094\u00E2\u0080\u00A2 k and subbing in, the result is l) \u00E2\u0080\u0094 T - i 1 v - ^ e 2iri(n\u00E2\u0080\u0094m)k/T k 02iri(n\u00E2\u0080\u0094m)k/T (A.29) T ^ | S i n 2 ( 7 r f c / T ) + a M 2 -This is the exact result for the 2-point correlation function that should appear in a Monte Carlo simulation of this problem. 78 Appendix A. Analytic Calculations for Matrix Integrals A.2 Matrix Harmonic Potential Next, consider a matrix harmonic potential with action 1 S e = 2J0 D T T V X2{t) + ^X\t) 2 y 2 / (A.30) and periodic boundary conditions X(t + 0) = X(t). Here the Xs are N x N Hermitian ma-trices. This system can easily be extended so that the Xs are in multiple spatial dimensions if desired. The partition function is Z = J[dX]e-y\u00C2\u00B0d^\x2^x2]. (A.31) To begin, the trace and integral measure are broken down into the components of the matrices t r ( X 2 ) = XabXba = 2 E | A a f e | 2 + E ( A a a ) 2 a***ab(8) \u00E2\u0080\u0094 e^9a~8b^ipab(0). The eigenfunctions satisfying this equation are (A.41) with eigenvalues \n \u00E2\u0080\u0094 Aab (2ixn | 9 a - + pi (A.42) Thus det dt2 ab n n=\u00E2\u0080\u0094oo L oo f2nn , 6 a - 9 b \ 2 , 2 - n /27rn v~7T n (2ixn V T Pi (A.43) where n+ = im + and ^ = ~ W + This product can be regularized by expressing it as an exponential and integrating over /j,', at which point it is expected to be determinable up to a constant: n (2im VT + P ; E J n ( ^ + / ) (A.44) This sum over n is actually a cotangent. To see this, consider the relation 7rcot(-7r^) = E z + n (A.45) Manipulating for z = gives that \u00E2\u0080\u009E ( l T + A*') ^ c o t f ^ 2 V 2 (A.46) 81 Appendix A. Analytic Calculations for Matrix Integrals and thus (A.47) = e In sin ( \u00C2\u00A3 f - ) + c i A * ' + C 2 Alternatively, this could be derived by matching the residues of equation (A.44) and per-forming a contour integration, but the details are not elaborated upon. Setting ci and c2 to zero, the partition function is Z = j d0i \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 d0jv JI s i n ( J a****eff = EE i=l a****\u00C2\u00AB-h)\ + l n s i n h + i(\u00C2\u00BB.-e>) Z Z I \ Z Z (A.50) ,-p~iii-i(6a-6 i=l a**** ( l - e - ^ + i ^ - \u00C2\u00B0 \u00C2\u00BB ) ) + N { N ~ l ) B p d . a^b In the last step all dimensions are taken to have the same mass parameter u. and the last term can be disregarded as a zero point energy. The final expression for integration over the matrices is then Z = f d6, \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 d6N H sin 2 f6\u00E2\u0080\u0094^) e - d ^ m ( i - e - ^ ^ - ^ ) . ( A . 5 1 ) \u00E2\u0080\u00A2* a****)) \u00C2\u00AB _e-/3/i+l(6,a-ei)) p^he partition function for this first order approximation is then Z \u00C2\u00AB IdO, \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 dQw J\"J sin 2 ( e * - ' \" E . , \u00C2\u00BB \u00C2\u00AB'<'-\">. (A.56) J a**** "Thesis/Dissertation"@en .
"10.14288/1.0084945"@en .
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"Finite temperature phase transitions in the quenched BFSS matrix model"@en .
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