The thermal state of a scalar field in quantum field theory (QFT) is a state in which the field is in equilibrium with a heat bath at a certain temperature. This state is described by a density matrix, which incorporates the notion of thermal fluctuations and represents the statistical distribution of the field's excitations.
The thermal state of a scalar field is different from other states in QFT because it is a mixed state, meaning that it is a combination of different pure states. In contrast, other states in QFT, such as the vacuum state, are pure states that are uniquely defined by a set of creation and annihilation operators.
The thermal state of a scalar field is determined by the temperature of the heat bath, the energy of the field's excitations, and the interactions between the field and the heat bath. These factors affect the statistical distribution of the field's excitations and determine the properties of the thermal state.
The thermal state of a scalar field is closely related to thermodynamics, as it represents a state of thermal equilibrium with a heat bath. The statistical distribution of the field's excitations in this state follows the laws of thermodynamics, and the field's energy and entropy can be calculated using thermodynamic principles.
The thermal state of a scalar field can be indirectly observed in experiments by measuring the effects of its excitations on other particles or fields. However, directly measuring the statistical distribution of the field's excitations in this state is challenging and requires specialized techniques such as quantum tomography.