In quantum field theory (QFT), the Lagrangian density is typically used as a fundamental starting point rather than the Hamiltonian, as is done in non-relativistic quantum mechanics. This choice is motivated by several reasons:
Lorentz Invariance: QFT is designed to incorporate both quantum mechanics and special relativity. The Lagrangian density is formulated in a Lorentz-invariant manner, meaning it is consistent with the principles of special relativity. Lorentz invariance ensures that physical laws and predictions remain the same for all observers in different inertial frames of reference. By using the Lagrangian density, QFT naturally respects this crucial symmetry.
Field Quantization: In QFT, particles are considered as excitations of underlying quantum fields. The Lagrangian density provides a straightforward framework for quantizing these fields. By treating the fields as operators, the Lagrangian density allows for the creation and annihilation of particles and describes their interactions. The quantization procedure based on the Lagrangian density leads to the construction of Fock space, where particles are associated with different excitation states of the fields.
Symmetries and Conservation Laws: The Lagrangian density formulation of QFT makes it easier to incorporate symmetries and conservation laws. Symmetries play a fundamental role in physics, and the Lagrangian density is designed to be invariant under specific symmetry transformations. Noether's theorem connects these symmetries to conserved quantities, such as energy, momentum, and angular momentum. The Lagrangian density formulation naturally allows for the identification and understanding of these conservation laws.
Path Integral Formulation: The Lagrangian density is intimately connected to the path integral formulation of QFT. The path integral provides a powerful framework for calculating probabilities and transition amplitudes in QFT. By expressing the Lagrangian density in terms of the fields and their derivatives, the path integral formulation allows for the calculation of transition amplitudes by summing over all possible field configurations.
While the Hamiltonian is the central operator in non-relativistic quantum mechanics, it is still present in QFT. In fact, it can be derived from the Lagrangian density using the Euler-Lagrange equations. However, the Lagrangian density provides a more intuitive and natural starting point for the formulation of QFT due to its Lorentz invariance, its connection to field quantization, and its ability to incorporate symmetries and conservation laws.
It is worth noting that different formalisms can be used in QFT, such as the canonical quantization approach based on the Hamiltonian. These alternative approaches can provide useful insights and different perspectives on specific aspects of QFT. However, the Lagrangian density formulation remains the most common and widely used starting point in modern QFT due to its compatibility with special relativity and its ability to handle field quantization and symmetries.