This report aims to give an insight to the path integral formalism in quantum mechanics. After explaining the kernel's construction, some of its properties and ways to compute it, we see how it relates to the Schrödinger picture. Moreover, we see how its representation can change if it is defined in the space, momentum, time or energy space. Finally, we derive Born's expansion with the kernel showing how this formalism helps to understand perturbation theory and thus scattering. The path integral formalism is then used in quantum field theory with proofs and examples of simple correlation functions. Furthermore, supersymmetry in zero and one dimension are studied with use of the localization principle and the Witten index.

The focus of this thesis is to introduce the path integral and some of its applications. One interpretation of quantum mechanics is that a microscopic system which moves from an initial- to a final state moves through each possible intermediate state. The path integral uses the principle of least action to sum over all such intermediate states to find the evolution of a quantum mechanical system. We compare the path integral approach to that of the Schrödinger equation and show that the two give an equivalent description of quantum mechanics. To demonstrate the usefulness of the path integral, we introduce low-dimensional quantum field theory (QFT). In particular, we discuss Feynman diagrams. The idea behind Feynman diagrams is to sum over all possible weak interactions between fields to evaluate the properties of a system through the path integral. We also carry out a computation of a low energy effective action in a 0-dimensional model. The result of the computation shows that there is free energy also in a vacuum. Finally, we briefly generalize some of the previous discussion to 1-dimensional QFT. To give an example of a practical application, we give a qualitative discussion of how the path integral can be applied to statistical mechanics to predict the behaviour of superfluids. / Målet med den här rapporten är att introducera konceptet vägintegral och några av dess applikationer. En tolkning av kvantmekanik är att ett mikroskopiskt system som går från ett initialt- till ett slutgiltigt tillstånd kommer att passera genom alla möjliga mellanliggande tillstånd. Vägintegralen använder sig av principen om minsta verkan för att summera över alla sådana mellanliggande tillstånd för att hitta utvecklingen hos ett system. Vi kommer att jämföra vägintegralen med Schrödingers ekvation och visa att de två ger en ekvivalent beskrivning av kvantmekaniken. För att demonstrera vägintegralens användbarhet introducerar vi lågdimensionell kvantfältteori. Vi diskuterar speciellt Feynmandiagram. Idén bakom Feynmandiagram är att summera över alla möjliga svaga interaktioner mellan fält för att utvärdera fysikaliska egenskaper hos system med hjälp av vägintegraler. Vi kommer också att utvärdera en effektiv verkan i 0-dimensionell kvantfältteori. Resultatet visar att det finns fri energi även i ett vakuum. Slutligen generaliserar vi delar av vår tidigare diskussion till 1-dimensionell kvantfältteori. Som ett exempel på praktiska applikationer för vi en kvalitativ diskussion kring hur vägintegraler kan användas inom statistisk mekanik för att förutsäga egenskaper hos superfluider.

This dissertation introduces advances in the Wiener path integral (WPI) technique for determining efficiently and accurately the stochastic response of diverse nonlinear dynamical systems. First, a novel, general, formalism of the WPI technique is developed to account, in a direct manner, also for systems with non-Markovian response processes. Specifically, the probability of a path and the associated transition probability density function (PDF) corresponding to the Wiener excitation process are considered. Next, a functional change of variables is employed, in conjunction with the governing stochastic differential equation, for deriving the system response joint transition PDF as a functional integral over the space of possible paths connecting the initial and final states of the response vector. In comparison to alternative derivations in the literature, the herein-developed formalism does not require the Markovian assumption for the system response process. Overall, the veracity and mathematical legitimacy of the WPI technique to treat also non-Markovian system response processes are demonstrated. In this regard, nonlinear systems with a history-dependent state, such as hysteretic structures or oscillators endowed with fractional derivative elements, can be accounted for in a direct manner—that is, without resorting to any ad hoc modifications of the WPI technique pertaining, typically, to employing additional auxiliary filter equations and state variables. Next, a reduced-order WPI formulation is introduced for efficiently determining the stochastic response of diverse nonlinear systems with fractional derivative elements. This formulation can be also construed as a dimension reduction approach that renders the associated computational cost independent of the total number of stochastic dimensions of the problem. In fact, the proposed technique can determine, directly, any lower-dimensional joint response PDF corresponding to a subset only of the response vector components. This is accomplished by utilizing an appropriate combination of fixed and free boundary conditions in the related variational, functional minimization, problem. Notably, the reduced-order WPI formulation is particularly advantageous for problems where the interest lies in, few only, specific degrees-of-freedom whose stochastic response is critical for the design and optimization of the overall system. Further, an extrapolation approach within the WPI technique is developed that significantly enhances the computational efficiency of the technique without, practically, affecting the associated degree of accuracy. Overall, the WPI technique treats the system response joint transition PDF as a functional integral over the space of all possible paths connecting the initial and the final states of the response vector. Next, the functional integral is evaluated, ordinarily, by considering the contribution only of the most probable path. This corresponds to an extremum of the functional integrand, and is determined by solving a functional minimization problem that takes the form of a deterministic boundary value problem (BVP). This BVP corresponds to a specific grid point of the response PDF domain. Remarkably, the BVPs corresponding to two neighboring grid points not only share the same equations, but also the boundary conditions differ only slightly. This unique aspect of the technique is exploited, and it is shown that solution of a BVP and determination of the response PDF value at a specific grid point can be used for extrapolating and estimating efficiently and accurately the PDF values at neighboring points without the need for considering additional BVPs. Last, a joint time-space extrapolation approach within WPI technique is developed for determining, efficiently and accurately, the non-stationary stochastic response of diverse nonlinear dynamical systems. The approach can be construed as an extension of the above space-domain extrapolation scheme to account also for the temporal dimension. Specifically, it is shown that information inherent in the time-history of an already determined most probable path can be used for evaluating points of the response PDF corresponding to arbitrary time instants, without the need for solving additional BVPs. In a nutshell, relying on the aforementioned unique and advantageous features of the WPI-based BVP, the complete non-stationary response joint PDF is determined, first, by calculating numerically a relatively small number of PDF points, and second, by extrapolating in the joint time-space domain at practically zero additional computational cost. Compared to an alternative brute-force implementation of the WPI technique, and to a standard Monte Carlo simulation (MCS) solution treatment, the developed extrapolation approach reduces the associated computational cost by several orders of magnitude. Several representative numerical examples are considered to demonstrate the reliability of the developed techniques. Juxtapositions with pertinent MCS data are included as well.

This thesis explores the theoretical representation of localised electrons in magnetic systems, using Majorana fermions. A motivation is provided for the Majorana fermion representation, which is then developed and applied as a mean-field theory and in the path-integral formalism to the Ising model in transversal-field (TFIM) in one, two and three dimensions, on an orthonormal lattice. In one dimension the development of domain walls precludes long-range order in discrete systems; this is as free energy savings due to entropy outweigh the energetic cost of a domain wall. An argument due to Peierls exists in 2D which allows the formation of domains of ordered spins amidst a disordered background, however, which may be extended to 3D. The forms of the couplings to the bosons used in the Random Phase Analysis (RPA) are considered and an explanation for the non-existence of the phases calculated in this thesis is discussed, in terms of spare degrees of freedom in the Majorana representation. This thesis contains the first known application of Majorana fermions at the RPA level.

In this thesis the path integral formalism is applied to the calculation of the dynamics of dissipative quantum systems. The time evolution of a system of bilinearly coupled bosonic modes is treated using the real-time path integral technique in coherent-state representation. This method is applied to a damped harmonic oscillator within the Caldeira-Leggett model. In order to get the stationary trajectories the corresponding Lagrangian function is diagonalized and then the path integrals are evaluated by means of the stationary-phase method. The time evolution of the reduced density matrix in the basis of coherent states is given in simple analytic form for weak system-bath coupling, i.e. the so-called rotating-wave terms can be evaluated exactly but the non-rotating-wave terms only in a perturbative manner. The validity range of the rotating-wave approximation is discussed from the viewpoint of spectral equations. In addition, it is shown that systems without initial system-bath correlations can exhibit initial jumps in the population dynamics even for rather weak dissipation. Only with initial correlations the classical trajectories for the system coordinate can be recovered. The path integral formalism in a combined phase-space and coherent-state representation is applied to the problem of curve-crossing dynamics. The system of interest is described by two coupled one-dimensional harmonic potential energy surfaces interacting with a heat bath. The mapping approach is used to rewrite the Lagrangian function of the electronic part of the system. Using the Feynman-Vernon influence-functional method the bath is eliminated whereas the non-Gaussian part of the path integral is treated using the perturbation theory in the small coordinate shift between potential energy surfaces. The vibrational and the population dynamics is considered in a lowest order of the perturbation. The dynamics of a Gaussian wave packet is analyzed along a one-dimensional reaction coordinate. Also the damping rate of coherence in the electronic part of the relevant system is evaluated within the ordinary and variational perturbation theory. The analytic expressions for the rate functions are obtained in the low and high temperature regimes.

This thesis presents the benchmarking and development of a method to study ground state properties of hydrogen clusters using molecular dynamics. Benchmark studies are performed on our Path Integral Molecular Dynamics code using the Langevin equation for finite temperature studies and our Langevin equation Path Integral Ground State code to study systems in the zero-temperature limit when all particles occupy their nuclear ground state. A simulation is run on the first 'real' system using this method, a parahydrogen molecule interacting with a fixed water molecule using a trivial unity trial wavefunction. We further develop a systematic method of optimizing the necessary parameters required for our ground state simulations and introduce more complex trial wavefunctions to study parahydrogen clusters and their isotopologues orthodeuterium and paratritium. The effect of energy convergence with parameters is observed using the trivial unity trial wavefunction, a Jastrow-type wavefunction that represents a liquid-like system, and a normal mode wavefunction that represents a solid-like system. Using a unity wavefunction gives slower energy convergence and is inefficient compared to the other two. Using the Lindemann criterion, the normal mode wavefunction acting on floppy systems introduces an ergodicity problem in our simulation, while the Jastrow does not. However, even for the most solid-like clusters, the Jastrow and the normal mode wavefunctions are equally efficient, therefore we choose the Jastrow trial wavefunction to look at properties of a range of cluster sizes. The energetic and structural properties obtained for parahydrogen and orthodeuterium clusters are consistent with previous studies, but to our knowledge, we may be the first to predict these properties for neutral paratritium clusters. The results of our ground state simulations of parahydrogen clusters, namely the distribution of pair distances, are used to calculate Raman vibrational shifts and compare to experiment. We investigate the accuracy of four interaction potentials over a range of cluster sizes and determine that, for the most part, the ab initio derived interaction potentials predict shifts more accurately than the empirically based potentials for cluster sizes smaller than the first solvation shell and the trend is reversed as the cluster size increases. This work can serve as a guide to simulate any system in the nuclear ground state using any trial wavefunction, in addition to providing several applications in using this ground state method.

Orientador: Pedro Jose Catuogno / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatística e Computação Cientifica / Made available in DSpace on 2018-08-14T14:39:55Z (GMT). No. of bitstreams: 1 Castrequini_RafaelAndretto_M.pdf: 934326 bytes, checksum: e4c45bc1efde09bbe52710c44eab8bbf (MD5) Previous issue date: 2009 / Resumo: Introduzimos a teoria dos p-rough paths seguindo a abordagem de M. Gubinelli, conhecida por integração algébrica. Durante toda a dissertação nos restringimos ao caso 1 </= p < 3, o que e suficiente para lidar com trajetórias do movimento Browniano e aplicações ao Cálculo Estocástico. Em seguida, estudamos as equações diferenciais associadas aos rough paths, onde nós conectamos a abordagem de A. M. Davie (as equações) e a abordagem de M. Gubinelli (as integrais). No final da dissertação, aplicamos a teoria de rough path ao cálculo estocástico, mais precisamente relacionando as integrais de Itô e Stratonovich com a integral ao longo de caminhos. / Abstract: We introduce p-Rough Path Theory following M. Gubinelli_s approach, as known as algebraic integration. Throughout this masters thesis, we are concerned only in the case where 1 </= p < 3, witch is enough to deal with trajectories of a Brownnian motion and some applications to Stochastic Calculus. Afterwards, we study differential equations related to rough paths, where we connect the approach of A. M. Davie to equations with the approach of M. Gubinelli to integrals. At the end of this work, we apply the theory of rough paths to stochastic calculus, more precisely, we related the integrals of Itô and Stratonovich to integral along paths. / Mestrado / Sistemas estocasticos / Mestre em Matemática

In this thesis we aim to test if Landau\'s two-component superfluid model is compatible with the quark-gluon-plasma description. We follow the test proposed by Chernodub et. al. [Two-component liquid model for the quark-gluon plasma. Theor. Math. Phys., v. 170, p. 211&ndash;216, 2012]. We start by reviewing the building process of a field theory with gauge symmetries and discussing the conservation laws associated to the theory&rsquo;s symmetries. We explore the thermodynamic approach to quantum theory and the interesting fact that, when combined with a field theory, the path-integral formulation for quantum field theories emerges naturally. We also present the necessity of introducing a momentum cutoff into the theory and show that embedding space-time on a lattice is a way to introduce this cutoff and renormalize the theory. As a bonus, this also allows the numerical and non-perturbative evaluation of observables. We overview the phenomenological aspects of relativistic heavy-ion collisions and Landau&rsquo;s two-component model for superfluids, along with a quantum-field-theory motivation for it, and explain details of the test proposed by Chernodub et. al.. Lastly, we show the implementation details of our simulation along with results. We do not see evidence that the proposed superfluid model is able to describe the plasma. We speculate that this might be caused by the absence of fermions in our simulations. / Nesta tese nosso objetivo é testar se o modelo de Landau de duas componentes para superfluidos é compatível com a descrição do plasma de quarks e glúons. Seguimos o teste proposto por Chernodub et. al. [Two-component liquid model for the quark-gluon plasma. Theor. Math. Phys., v. 170, p. 211&ndash;216, 2012]. Começamos revisando o processo de construção de uma teoria de campo com simetria de gauge e discutindo as leis de conservação associadas às simetrias da teoria. Exploramos a abordagem termodinâmica para teoria quântica e o interessante fato de que, quando combinada com uma teoria de campo, a formulação de integrais de trajetória para teorias quânticas de campo emerge naturalmente. Também apresentamos a necessidade de se introduzir um corte de momento na teoria, e mostramos que embutir o espaço-tempo em uma rede é um meio de introduzir o corte na teoria e renormalizá-la. Como um bônus, isso também permite o cálculo numérico e não-perturbativo de observáveis. Apresentamos um panorama dos aspectos fenomenológicos da colisão de íons pesados relativísticos e o modelo de duas componentes de Landau para superfluidos, bem como uma motivação de teoria quântica de campo para ele, e explicamos detalhes do teste proposto por Chernodub et. al.. Por fim, mostramos os detalhes de nossa implementação juntamente com nossos resultados. Não vemos evidência de que o modelo de superfluidod proposto seja capaz de descrever o plasma. Nós especulamos que isto possa ser causado pela ausência de férmions em nossas simulações.

Orientador: Amir Ordacgi Caldeira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Fisica Gleb Wataghin / Made available in DSpace on 2018-08-18T12:04:56Z (GMT). No. of bitstreams: 1 GarciaRodriguez_AlexisOmar_D.pdf: 2441638 bytes, checksum: 00dfa78fb7b0c9f69778a51704c587b7 (MD5) Previous issue date: 2011 / Resumo: Este trabalho foi motivado por várias dificuldades encontradas no estudo do artigo de M. Ueda, Transmission Spectrum of a Tunneling Particle Interacting with Dynamical Fields: Real- Time Functional Integral Approach, Phys. Rev. B 54, 8676 (1996). Nesse artigo, num formalismo de tempo real, é descrito o tunelamento de uma partícula através de uma barreira utilizando tempos não reais de travessia através dessa barreira. No presente trabalho é proposto um formalismo mais amplo de tempo real para uma introdução mais natural de valores complexos do tempo na descrição do tunelamento de uma partícula cm interação com o ambiente. Esta proposta está baseada no chamado método do tempo complexo utilizado no caso do tunelamento de uma partícula sem interação com o ambiente estudado nos trabalhos de D. W. McLaughlin, J. Math. Phys. 13, 1099 (1972) c B. R. Holstein c A. R. Swift, Am. J. Phys. 50, 833 (1982). Seguindo o trabalho citado de Ueda, o ambiente da partícula é representado através de um conjunto, ou banho térmico, de osciladores harmônicos caracterizados por uma função de densidade espectral J(w). Utilizando o método de Feynman de integrais de trajetória, integramos sobre as coordenadas dos osciladores do banho c obtemos uma expressão exata para o espectro de transmissão da partícula para uma temperatura do banho T > O. Limitando-nos então ao caso mais simples T = O, estudamos o tunelamento dissipativo da partícula através da barreira. Considerando h um parâmetro pequeno (limite semiclássico), aproximamos o espectro de transmissão da partícula através da contribuição das trajetórias clássicas c suas trajetórias vizinhas. Nesta aproximação consideramos a variação da ação efetiva da partícula para tempos dados de duração das trajetórias c deste modo substituímos o procedimento variacional seguido no trabalho indicado de Ueda onde não é considerada a variação nos tempos de travessia da partícula através da barreira. Num segundo problema variacional nos tempos de duração das trajetórias clássicas de acordo com o método do tempo complexo e considerando também a variação nas posições iniciais c finais dessas trajetórias, obtemos as equações de movimento das chamadas trajetórias clássicas especiais. Este tratamento das coordenadas iniciais c finais das trajetórias clássicas substitui o procedimento seguido no trabalho de Ueda onde é considerc1da uma aceleração nula durante todo o trajeto de movimento incluindo o trajeto na região da barreira. Diferentemente do artigo citado de Ueda, no presente trabalho utilizamos pacotes de ondas relativamente bem localizados para descrever os estados inicial e final da partícula. Em consequência, aproximamos o espectro de transmissão da partícula através de trajetórias clássicas especiais com coordenadas iniciais c finais iguais ao valor médio da coordenada para esses pacotes de ondas. O procedimento seguido neste trabalho, baseado no método do tempo complexo, permite obter o fator ele acoplamento apropriado entre as duas trajetórias que descrevem a ação efetiva ela partícula substituindo assim o procedimento de tipo ad hoc seguido com este fim no trabalho indicado de Ueda. O método do tempo complexo permite obter também o termo ela diferença entre a ação efetiva da partícula c o expoente ele tunelamento, sendo que estas grandezas são tratadas como iguais no trabalho citado de Ueda. Considerando termos até primeira ordem num campo elétrico externo c na interação da partícula com o banho de osciladores, obtemos expressões gerais para o expoente de tunelamento, o espectro de transmissão, a taxa total de tunelamento c o tempo de travessia da partícula através da barreira, válidas para um banho de osciladores com uma função de densidade espectral arbitrária. Assim temos que a interação da partícula com um banho de osciladores com uma função de densidade espectral arbitrária diminui a taxa total de tunelamento. Adicionalmente, obtemos que a interação da partícula com os osciladores do banho com frequências ?a = ?C ~ 1.9 T , onde T0 é o tempo característico de travessia através da barreira no caso cm que não há interação da partícula com o banho de osciladores nem campo elétrico, não afeta o tempo característico de travessia através da barreira. Por outro lado, a interação da partícula com os osciladores do banho que têm frequências ?a < ?C (?a > ?C) diminui (aumenta) o tempo característico de travessia através da barreira. No caso de um banho de osciladores com uma única frequência w c uma constante de acoplamento com a partícula dada por Ca = Ca (wT)a , são identificados cinco comportamentos diferentes em função de w para o expoente característico de tunelamento e o tempo característico de travessia através da barreira. Estes comportamentos correspondem aos valores de s < 1, s = 1, 1 < s < 2, s = 2 e s > 2. No trabalho de M. Ueda, Phys. Rev. B 54, 8676 (1996), foi considerado somente o expoente característico de tunelamento no caso s = 1. No caso de um banho ôhmico de osciladores a temperatura zero, assim corno no caso de um banho de osciladores com uma única frequência, obtemos que o espectro de transmissão da partícula é zero para urna energia final característica da partícula maior que a energia inicial característica. Este resultado corrige o resultado correspondente no trabalho citado de Ueda, o qual não é consistente do ponto de vista físico, permitindo também obter de um modo mais coerente a corrente de tunelamento entre dois metais separados por um material isolante a temperatura zero. Obtém-se também que a interação da partícula com um banho ôhmico de osciladora não afeta o tempo característico de travessia através da barreira até primeira ordem nessa interação / Abstract: This work was motivated by several difficulties found when studying the article by M . Ueda, Transmission Spectrum of a Tunneling Particle Interacting with Dynamical Fields: Real-Time Functional-Integral Approach, Phys. Rev. B 54, 8676 (1996). In that paper, using a real-time formalism, a tunneling particle is described by complex traversal times of tunneling. In the present work we propose a broader real-time formalism that allows for a more natural introduction of complex values of time in the description of a tunneling particle interacting with the environment. This proposal is based on the well-known complex time method used in the case of a tunneling particle with no interaction with the environment studied in the works of D. W. McLaughlin, J. Math. Phys. 13, 1099 (1972) and B. R. Holstein and A. R. Swift, Am. J. Phys. 50, 833 (1982). Following the cited work of Ueda, the environment of the particle is represented by a set, or heat bath, of harmonic oscillators which is characterized by a spectral density function J(w). Using the Feynman path integrals method, we integrate out the coordinates of the bath oscillators and obtain an exact expression for the transmission spectrum of the particle for a bath temperature T > O. Limiting ourselves to the simpler case T = O, we study the case of a dissipative tunneling of the particle. Considering h a small parameter (semiclassical limit) we approximate the transmission spectrum of the particle by the contribution of the classical trajectories and its neighboring paths. In this approach we consider the variation of the effective action of the particle for given duration times of the paths and replace the variation procedure followed in the cited work of Ueda where the variation in the traversal times of tunneling is not considered. In a second variation problem for the duration times of the classical paths, according to the complex time method and considering also the variation in the initial and final positions of these paths, we obtain the equations of motion for the so-called special classical paths. This treatment of the initial and final coordinates of the classical paths replaces the procedure followed in the cited work of Ueda where an acceleration equal to zero is considered during the entire path of motion including the region under the barrier. Unlike the cited article of Ueda, we use in the present work wave packets relatively well localized to describe the init.ial and final statics of the particle. Conscqncnt.ly, we approximate the transmission spectrum of the particle through special classical paths with initial and final coordinates equal to the average value of the coordinate for those wave packets. The procedure followed in this work, based on the complex time method, gives the appropriate coupling factor between the two paths describing the effective action of the particle and thus replaces the ad hoc procedure followed for this purpose in the cited work of Ueda. The complex time method also allows us to obtain the difference term between the effective action of the particle and the tunneling exponent. These quantities are treated as equal in Ueda\'s work. Considering terms up to first order in an external electric field and the interaction of the particle with the bath of oscillators, we obtain general expressions for the tunneling exponent, transmission spectrum, total tunneling rate and traversal time of tunneling, which are valid for a bath of oscillators with an arbitrary spectral lenity function. We find that the interaction of the particle with a bath of oscillators with an arbitrary spectral density function decreases the total tunneling rate. Also, we find that the interaction of the particle with the bath oscillators with frequencies ?a = ?C ~ 1.9 T , where To is the characteristic traversal time of tunneling when there is no interaction of the particle with the bath of oscillators nor electric field. , does not affect the characteristic traversal time of tunneling. On the other hand, the interaction of the particle with the bath oscillators having frequencies ?a< ?c (?a: > ?c decreases (increases) the characteristic traversal time of tunneling. In the case of a bath of oscillators with a single frequency w and a coupling constant with the particle given by Ca = Ca (wT)a we identify five different behaviors deepening on w for the characteristic tunneling exponent and the characteristic traversal time of tunneling. These behaviors correspond to the values of s < 1, s = 1, 1 < s < 2, s = 2 and s > 2. In the work of M. Ueda, Phys. Rev. B 54, 8676 (1996), it was only considered the characteristic tunneling exponent in the case s = 1. In the case of an ohmic bath of oscillators at zero temperature, as well as in the case of a bath of oscillators with a single frequency, we obtain that the transmission spectrum of the particle is ;1,cro for a final characteristic energy of the particle greater than the initial characteristic energy. This result corrects the corresponding result in Ueda work, which is not consistent from a physical point of view, allowing also for a more coherent derivation of the tunneling current between two metals separated by an insulating material at zero temperature. It is also obtained that the interaction of the particle with an ohmic bath of oscillators does not affect the characteristic traversal time of tunneling up to first order in that interaction / Doutorado / Física / Doutor em Ciências

In this thesis the path integral formalism is applied to the calculation of the dynamics of dissipative quantum systems. The time evolution of a system of bilinearly coupled bosonic modes is treated using the real-time path integral technique in coherent-state representation. This method is applied to a damped harmonic oscillator within the Caldeira-Leggett model. In order to get the stationary trajectories the corresponding Lagrangian function is diagonalized and then the path integrals are evaluated by means of the stationary-phase method. The time evolution of the reduced density matrix in the basis of coherent states is given in simple analytic form for weak system-bath coupling, i.e. the so-called rotating-wave terms can be evaluated exactly but the non-rotating-wave terms only in a perturbative manner. The validity range of the rotating-wave approximation is discussed from the viewpoint of spectral equations. In addition, it is shown that systems without initial system-bath correlations can exhibit initial jumps in the population dynamics even for rather weak dissipation. Only with initial correlations the classical trajectories for the system coordinate can be recovered. The path integral formalism in a combined phase-space and coherent-state representation is applied to the problem of curve-crossing dynamics. The system of interest is described by two coupled one-dimensional harmonic potential energy surfaces interacting with a heat bath. The mapping approach is used to rewrite the Lagrangian function of the electronic part of the system. Using the Feynman-Vernon influence-functional method the bath is eliminated whereas the non-Gaussian part of the path integral is treated using the perturbation theory in the small coordinate shift between potential energy surfaces. The vibrational and the population dynamics is considered in a lowest order of the perturbation. The dynamics of a Gaussian wave packet is analyzed along a one-dimensional reaction coordinate. Also the damping rate of coherence in the electronic part of the relevant system is evaluated within the ordinary and variational perturbation theory. The analytic expressions for the rate functions are obtained in the low and high temperature regimes.